151 Plane Geometry Terms and Definitions | Mathematics Board Exam Review

INTRODUCTION

Plane geometry is the study of figures that lie entirely in a flat surface namely points, lines, angles, triangles, circles, and polygons and the relationships between them. For Philippine engineering licensure examinees, it is one of the most reliably tested mathematics subjects across all engineering programs. The problems are not usually the hardest on the exam, but they are the ones that cost reviewees the most points simply because the vocabulary and the theorems were never fully learned in the first place. Knowing the terms cold is what converts a geometry problem from intimidating to straightforward.

This expanded reference covers 151 terms drawn from the complete scope of Plane Geometry as tested in the PRC examinations. The list has been significantly expanded from the standard 101-term coverage to include classical triangle theorems such as Ceva’s theorem, Menelaus’ theorem, Stewart’s theorem, and the Apollonius theorem for median length, as well as a full treatment of triangle centers, circle power theorems, locus definitions, and the properties of cyclic polygons. Each definition is written with the board exam in mind and the formula, the geometric meaning, and the problem context are all included wherever they are relevant.

Alphabetical ordering is used throughout so you can locate any term quickly during your review sessions. As you read, connect each term to the cluster it belongs to such as triangle theorems, circle theorems, polygon properties, or locus and transformation. Plane geometry is built on a small number of deep ideas that generate a large number of specific results. If you understand the ideas, the specific results follow. If you only memorize the results without understanding, a slightly unfamiliar problem will stop you cold.

Use this list as your primary reference for Plane Geometry throughout your PRC review cycle. Flag the terms you are uncertain about, revisit them with worked examples, and pay particular attention to the circle angle theorems, the triangle center properties, and the classical theorems in the second half of the list. Those are the terms that separate average scores from high ones on the board exam.

The 101 Plane Geometry Terms and Definitions

1. Acute Angle

An angle measuring greater than 0° but less than 90°. Acute angles appear in most triangle problems and in circle problems where arcs smaller than a semicircle are involved. Recognizing whether an angle is acute helps determine which trigonometric form of the area or law formula is appropriate.

2. Acute Triangle

A triangle in which all three interior angles are less than 90°. The orthocenter of an acute triangle lies inside the triangle. The circumcenter also lies inside the triangle. Board problems involving altitudes, medians, and the circumscribed circle frequently involve acute triangles.

3. Adjacent Angles

Two angles sharing a common vertex and a common side with non-overlapping interiors. Adjacent angles are foundational in problems involving angle sums, linear pairs, and interior angle calculations for polygons. A linear pair is a special case of adjacent angles whose non-common sides form a straight line.

4. Altitude of a Triangle

A perpendicular segment from a vertex to the line containing the opposite side. Every triangle has three altitudes meeting at the orthocenter. The altitude is essential in the area formula A = ½ × base × height and in the geometric mean relations for right triangles where the altitude is drawn to the hypotenuse.

5. Angle

The figure formed by two rays sharing a common endpoint called the vertex. Angles are measured in degrees or radians. They are classified as acute (0° to 90°), right (90°), obtuse (90° to 180°), straight (180°), or reflex (180° to 360°). Angle relationships are the foundation of nearly every plane geometry theorem.

6. Angle Between Two Secants from an External Point

When two secants are drawn from an external point, the angle at the external point equals half the positive difference of the intercepted arcs. If the near arc is x° and the far arc is y°, then the angle is (y − x)/2. This is one of three configurations of angles formed outside a circle, along with the tangent-secant and tangent-tangent angles, all sharing the same half-difference formula.

7. Angle Bisector

A ray dividing an angle into two congruent parts. The three angle bisectors of a triangle meet at the incenter. The angle bisector length from vertex A to side BC can be computed using the formula t_a = (2bc cos(A/2)) / (b + c). The angle bisector is also used to locate the incenter and to apply the angle bisector theorem.

8. Angle Bisector Length Formula

For a triangle with sides a, b, c opposite to vertices A, B, C, the length of the bisector from vertex A to side BC is given by t_a = √(bc[(b+c)² − a²]) / (b+c). This formula is used in board problems that ask for the length of an angle bisector when all three sides are known. It is distinct from the median and altitude length formulas.

9. Angle Bisector Theorem

In a triangle, the bisector from vertex A to side BC divides BC into segments BD and DC satisfying BD/DC = AB/AC = c/b. The bisector divides the opposite side in the ratio of the two sides adjacent to the bisected angle. This theorem is used in problems where a side is divided internally in a given ratio by an angle bisector.

10. Angle in a Semicircle

Any inscribed angle subtending a diameter is exactly 90°. If one side of an inscribed angle is a diameter of the circle, the angle is always a right angle regardless of where the vertex sits on the circle. This theorem is one of the most directly tested circle theorems on the board exam and is the basis for identifying right triangles inscribed in circles.

11. Apothem

The perpendicular distance from the center of a regular polygon to the midpoint of any side. The apothem equals the inradius of the regular polygon. It is used in the area formula A = ½ × perimeter × apothem and in the formula A = (ns²) / (4 tan(π/n)). The apothem of a regular hexagon with side s equals s√3/2.

12. Apothem-Radius Relationship

For a regular n-gon with circumradius R (radius of circumscribed circle) and apothem a (radius of inscribed circle), the relationship is a = R cos(π/n). The side length is s = 2R sin(π/n). These two formulas connect the three key measurements of a regular polygon namely side, apothem, and circumradius and are used to compute area in the form A = ½nsa = (nR²/2) sin(2π/n).

13. Arc

A connected portion of a circle’s circumference. A minor arc subtends a central angle less than 180°. A major arc subtends a central angle greater than 180°. A semicircle is half the circle. Arc measure in degrees equals the central angle that intercepts it, which is the foundation of all circle angle theorems.

14. Arc Length

The distance along the curved path of an arc. For radius r and central angle θ in radians, arc length s = rθ. In degrees, s = (θ/360) × 2πr. Board problems ask for arc length given radius and central angle, or ask for the central angle given the arc length and radius. Arc length is also the integrand in the calculus formula for curve length.

15. Area of a Circle

The area enclosed by a circle of radius r, given by A = πr². In terms of diameter d, A = πd²/4. This formula appears in sector, segment, and composite area problems throughout the board exam. The area of an annulus (ring) between two concentric circles of radii r₁ and r₂ is π(r₂² − r₁²).

16. Area of a Cyclic Polygon

A polygon inscribed in a circle has its area related to its side lengths through generalized formulas. For a cyclic quadrilateral, the area is given by Brahmagupta’s formula. For a cyclic triangle, it is given by Heron’s formula. The area of any regular polygon inscribed in a circle of radius R is A = (nR²/2) sin(2π/n).

17. Area of a Rectangle

Computed as A = length × width = lw. The diagonal of a rectangle with sides l and w is d = √(l² + w²). Board problems often use rectangles in composite figures or in optimization problems where the perimeter is fixed and the area is to be maximized, giving l = w (the square gives maximum area for a fixed perimeter).

18. Area of a Regular Polygon

Given by A = ½ × perimeter × apothem = (ns²)/(4 tan(π/n)), where n is the number of sides and s is the side length. Alternatively, A = (nR²/2) sin(2π/n) where R is the circumradius. For common polygons: equilateral triangle A = (√3/4)s², square A = s², regular hexagon A = (3√3/2)s².

19. Area of a Trapezoid

Given by A = ½(b₁ + b₂)h, where b₁ and b₂ are the parallel bases and h is the perpendicular height. For an isosceles trapezoid, the area can also be computed from the diagonals and the angle between them. The midsegment of a trapezoid has length (b₁ + b₂)/2 and the area equals midsegment × height.

20. Area of a Triangle

The primary formula is A = ½ × base × height. When the height is unknown, Heron’s formula A = √[s(s−a)(s−b)(s−c)] applies, where s is the semi-perimeter. When two sides and the included angle are known, A = ½ab sin C. When the circumradius R is known, A = abc/(4R). All four forms are tested on the board exam.

21. Bisector

A point, line, ray, or segment that divides a geometric figure into two congruent parts. A perpendicular bisector divides a segment at its midpoint at a right angle. An angle bisector divides an angle into two equal parts. Bisectors are used to locate the circumcenter (perpendicular bisectors) and incenter (angle bisectors) of a triangle.

22. Brahmagupta’s Formula

The area of a cyclic quadrilateral with consecutive sides a, b, c, d is A = √[(s−a)(s−b)(s−c)(s−d)], where s = (a+b+c+d)/2 is the semi-perimeter. Brahmagupta’s formula is the direct generalization of Heron’s formula from triangles to cyclic quadrilaterals. It appears in board problems involving quadrilaterals inscribed in circles.

23. Central Angle

An angle whose vertex is at the center of a circle and whose sides pass through two points on the circle. The central angle equals the arc it intercepts in degree measure. The central angle is twice the inscribed angle subtending the same arc; this relationship is the foundation of the inscribed angle theorem and all its consequences.

24. Centroid

The point where the three medians of a triangle intersect. The centroid divides each median in the ratio 2:1 from vertex to midpoint of the opposite side. It is the center of mass (balance point) of a uniform triangular lamina. The centroid always lies inside the triangle and is located at the average of the three vertex coordinates: G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3).

25. Centroid Coordinates

The centroid of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) has coordinates G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3). This formula is the arithmetic mean of the vertex coordinates. It is used in coordinate geometry problems involving the centroid and in physics problems where the centroid gives the center of mass of a uniform triangular plate.

26. Ceva’s Theorem

In a triangle ABC, three cevians AD, BE, and CF (where D is on BC, E is on CA, and F is on AB) are concurrent if and only if (BD/DC)(CE/EA)(AF/FB) = 1. Ceva’s theorem is used to prove that the medians, angle bisectors, and altitudes each meet at a single point and to solve problems involving ratios of divided sides. It is the concurrency criterion for cevians.

27. Cevian

A line segment from a vertex of a triangle to a point on the opposite side (or its extension). Medians, altitudes, and angle bisectors are all special cevians. The general properties of cevians are governed by Ceva’s theorem (for concurrent cevians) and Stewart’s theorem (for cevian length). Understanding cevians unifies the study of all three special lines through a vertex.

28. Chord

A line segment with both endpoints on a circle. The diameter is the longest chord. The perpendicular from the center to a chord bisects the chord. Equal chords are equidistant from the center. The chord-chord angle theorem states that the angle formed by two intersecting chords equals half the sum of the two intercepted arcs.

29. Chord-Chord Angle Theorem

When two chords intersect inside a circle, the measure of the angle formed equals half the sum of the two arcs intercepted by the angle and its vertical angle. If two chords AC and BD intersect at point P inside the circle, then angle APB = ½(arc AB + arc CD). This is one of the three main circle angle theorems tested on the board exam.

30. Chord-Chord Power Theorem

When two chords intersect inside a circle at point P, the products of their segments are equal: PA × PB = PC × PD, where AB and CD are the two chords. This is a special case of the power of a point theorem for interior points. It is used to find unknown chord segment lengths when the other segments are given.

31. Chord Distance from Center

The perpendicular distance d from the center of a circle to a chord of length l satisfies l = 2√(r² − d²), or equivalently d = √(r² − (l/2)²). Equal chords are equidistant from the center. Longer chords are closer to the center. The distance-chord relationship is used in board problems involving two chords of different lengths and their distances from the center.

32. Circumcenter

The point where the three perpendicular bisectors of the sides of a triangle intersect. The circumcenter is equidistant from all three vertices and is the center of the circumscribed circle (circumcircle). It lies inside acute triangles, on the hypotenuse of right triangles, and outside obtuse triangles. The circumradius R = abc/(4A) where a, b, c are the sides and A is the area.

33. Circumference

The total length of a circle’s boundary, given by C = 2πr = πd. The circumference is the perimeter of a circle. It is used in arc length calculations, in problems involving rolling circles, and in the relationship between arc length and central angle. The ratio C/d = π is the definition of π.

34. Circumradius Formula

The circumradius of a triangle is R = abc/(4A) = a/(2 sin A) = b/(2 sin B) = c/(2 sin C), where a, b, c are the sides and A is the area. The formula R = a/(2 sin A) is the law of sines applied to the circumcircle. For a right triangle, R = c/2 where c is the hypotenuse. The circumradius appears in problems linking the circumcircle to the sides and angles of the triangle.

35. Circumscribed Circle (Circumcircle)

A circle passing through all three vertices of a triangle. Every triangle has a unique circumcircle. The center is the circumcenter and the radius is the circumradius R = abc/(4A) = a/(2 sin A). For a right triangle, the circumradius equals half the hypotenuse. Board problems ask for R given the sides or angles of the triangle.

36. Circumscribed Polygon

A polygon whose sides are all tangent to an inscribed circle. The circle inside is called the incircle. For a triangle circumscribing a circle, the inradius r = A/s where A is the area and s is the semi-perimeter. A quadrilateral can be circumscribed about a circle if and only if the sums of opposite sides are equal: a + c = b + d.

37. Collinear Points

Three or more points lying on the same straight line. Collinearity is verified using the slope test, the area-of-triangle test (zero area means collinear), or Menelaus’ theorem for points on the sides of a triangle. The collinearity condition is used in locus problems and in problems where three apparently separate conditions reduce to a single geometric constraint.

38. Complementary Angles

Two angles whose measures sum to exactly 90°. Each is the complement of the other. The two acute angles of a right triangle are always complementary. Complementary angle relationships appear in problems involving right triangles, perpendicular lines, and angle calculations in polygons with right angles.

39. Composite Figure

A plane figure made up of two or more basic geometric shapes. The area of a composite figure is found by adding the areas of the component shapes or by subtracting areas when one shape is cut from another. Board exam problems involving shaded regions are almost always composite figure problems requiring careful identification and combination of basic area formulas.

40. Concave Polygon

A polygon with at least one interior angle greater than 180° or a reflex angle. Also called a non-convex polygon. At least one diagonal of a concave polygon lies outside the polygon. The standard area and diagonal count formulas apply to concave polygons, but some theorems (such as those requiring all diagonals to be interior) do not.

41. Concurrent Lines

Three or more lines passing through a single common point. The medians of a triangle are concurrent at the centroid, the altitudes at the orthocenter, the angle bisectors at the incenter, and the perpendicular bisectors at the circumcenter. Concurrency of cevians is characterized by Ceva’s theorem. Proving concurrency is a classic board exam problem type.

42. Congruent Figures

Figures with exactly the same shape and size. Two triangles are congruent if they satisfy SSS, SAS, ASA, AAS, or HL (right triangles). Congruent figures have equal corresponding sides and equal corresponding angles. Congruence is the strongest form of similarity (scale factor = 1) and is used in proofs and in problems where equal lengths or angles must be established.

43. Convex Polygon

A polygon in which all interior angles are less than 180° and every diagonal lies inside the polygon. Any line segment connecting two interior points of a convex polygon lies entirely inside the polygon. The sum of interior angles of a convex n-gon is (n−2) × 180°. Regular polygons are always convex.

44. Cyclic Polygon

A polygon whose vertices all lie on a single circle (circumscribed circle). Every triangle is cyclic. For a quadrilateral to be cyclic, opposite angles must be supplementary. Brahmagupta’s formula gives the area of a cyclic quadrilateral. The diagonal formulas for cyclic quadrilaterals are governed by Ptolemy’s theorem.

45. Cyclic Quadrilateral

A quadrilateral inscribed in a circle. Its opposite angles are supplementary (sum to 180°). The area is given by Brahmagupta’s formula A = √[(s−a)(s−b)(s−c)(s−d)]. The diagonals satisfy Ptolemy’s theorem: p₁p₂ = ab + cd, where p₁ and p₂ are the diagonals and a, b, c, d are consecutive sides. These properties are directly tested on the board exam.

46. Diagonal of a Polygon

A line segment connecting two non-adjacent vertices of a polygon. A convex polygon with n sides has n(n−3)/2 diagonals. A quadrilateral has 2 diagonals, a pentagon has 5, a hexagon has 9. The diagonal count formula is a standard board exam formula. For a rectangle, both diagonals are equal and bisect each other.

47. Diagonal of a Rectangle

The diagonal of a rectangle with sides l and w is d = √(l² + w²). Both diagonals are equal and bisect each other. The angle θ that the diagonal makes with the longer side satisfies tan θ = w/l. Board problems on rectangles often ask for the diagonal given the sides, or for a side given the diagonal and the other side, which requires a direct application of the Pythagorean theorem.

48. Diagonal of a Square

The diagonal of a square with side s is d = s√2. Conversely, the side is s = d/√2 = d√2/2 given the diagonal. The two diagonals of a square are equal, bisect each other at right angles, and each bisects two vertex angles (giving 45° at each vertex). The diagonal formula is derived from the Pythagorean theorem applied to the 45-45-90 triangle formed by two sides and the diagonal.

49. Diameter

The longest chord of a circle, passing through the center. Its length is d = 2r. The diameter divides the circle into two semicircles. Any angle inscribed in a semicircle (with the diameter as one side) is a right angle. In board problems, the diameter is often the given quantity from which the radius, area, and circumference are derived.

50. Equal Chords and Equal Arcs

In the same circle or in equal circles, equal chords subtend equal arcs and are equidistant from the center. Conversely, equal arcs have equal chords. This theorem is used in problems where symmetry arguments are made about chords or arcs without explicitly computing lengths. It follows from the SAS congruence of the triangles formed by equal chords and their radii.

51. Equilateral Triangle

A triangle with all three sides equal and all three angles equal to 60°. The area is A = (√3/4)s², the height is h = (√3/2)s, the inradius is r = s/(2√3), and the circumradius is R = s/√3. The centroid, circumcenter, incenter, and orthocenter all coincide at the same point. Equilateral triangle problems are among the most commonly tested on the board exam.

52. Euler Line

The straight line passing through the orthocenter (H), centroid (G), and circumcenter (O) of any triangle. These three triangle centers are always collinear. The centroid divides the segment OH in the ratio OG:GH = 1:2. The Euler line does not exist for equilateral triangles because all centers coincide. Knowing the Euler line eliminates the need to locate each center independently when one is known.

53. Euler’s Formula for Polyhedra

For any convex polyhedron, V − E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces. In plane geometry, the analogous formula for a connected planar graph is V − E + F = 2. Euler’s formula is a topological result that connects the combinatorial structure of a geometric figure and appears in advanced board problems on networks and tessellations.

54. Excircle (Escribed Circle)

A circle tangent to one side of a triangle and to the extensions of the other two sides. Every triangle has three excircles, one opposite each vertex. The radius of the excircle opposite vertex A is rₐ = A/(s−a), where A is the triangle area and s is the semi-perimeter. Excircles appear in advanced triangle problems involving tangent lengths and mixed inradius-circumradius relationships.

55. Excircle Radius Formula

The radius of the excircle opposite vertex A is rₐ = A/(s−a), where A is the triangle area and s is the semi-perimeter. Similarly rᵦ = A/(s−b) and r꜀ = A/(s−c). The three excircle radii together with the inradius satisfy the identity 1/r = 1/rₐ + 1/rᵦ + 1/r꜀ for certain triangle configurations. Excircle formulas appear in advanced triangle problems on the board exam.

56. Exterior Angle of a Polygon

The angle formed between one side of a polygon and the extension of an adjacent side. The sum of all exterior angles of any convex polygon, one at each vertex, is always 360°. Each exterior angle of a regular n-gon equals 360°/n. The exterior angle theorem for triangles states that an exterior angle equals the sum of the two non-adjacent interior angles.

57. Exterior Angle Theorem

The exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles. If the exterior angle is ∠ACD, then ∠ACD = ∠A + ∠B. This theorem is used to find unknown angles quickly without solving a full equation and appears in multi-step angle calculation problems throughout the board exam.

58. External Tangent Length

The length of a tangent segment from an external point P to a circle of radius r with center O at distance d is t = √(d² − r²). This follows directly from the Pythagorean theorem applied to the right triangle formed by the tangent, the radius, and the line from P to O. The tangent length formula is used in all problems involving tangent lines from external points.

59. Geometric Mean in Right Triangles

When the altitude is drawn from the right angle to the hypotenuse of a right triangle, three geometric mean relationships hold: the altitude is the geometric mean of the two hypotenuse segments (h² = pq), each leg is the geometric mean of the hypotenuse and the adjacent segment (a² = cp and b² = cq). These are used in problems where the altitude to the hypotenuse creates two smaller similar triangles.

60. Heron’s Formula

The area of a triangle with sides a, b, c is A = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2 is the semi-perimeter. It is used when all three sides are known and no height is given. Heron’s formula is one of the most frequently tested area formulas on the board exam and is the foundation for Brahmagupta’s formula for cyclic quadrilaterals.

61. Hexagon — Regular

A regular hexagon with side s has area A = (3√3/2)s², perimeter 6s, and can be divided into six equilateral triangles. The circumradius equals the side length: R = s. The apothem is a = s√3/2. The diagonals have lengths s√3 (short diagonal) and 2s (long diagonal). Regular hexagon properties appear frequently on the board exam because of their relation to equilateral triangles.

62. Hypotenuse

The side of a right triangle opposite the right angle. It is the longest side and satisfies the Pythagorean theorem c² = a² + b². The circumradius of a right triangle equals half the hypotenuse: R = c/2. In problems involving 30-60-90 or 45-45-90 triangles, the hypotenuse is typically the starting dimension from which the legs are derived.

63. Incenter

The point where the three angle bisectors of a triangle meet. The incenter is equidistant from all three sides and is the center of the inscribed circle. It always lies inside the triangle. The inradius is r = A/s, where A is the area and s is the semi-perimeter. The incenter coordinates are the weighted average of the vertices using side lengths as weights.

64. Incircle-Excircle Identity

For a triangle with inradius r, circumradius R, and semi-perimeter s, the area satisfies A = rs. This gives a direct link between the incircle and the triangle dimensions. The identity r = (s−a) tan(A/2) connects the inradius to the half-angles. The product of the exradii satisfies rₐrᵦr꜀ = r × s², connecting the incircle and excircles through the semi-perimeter.

65. Inradius Formula

The inradius of a triangle is r = A/s, where A is the area and s = (a+b+c)/2 is the semi-perimeter. Alternatively, r = (s−a) tan(A/2). For an equilateral triangle with side s, r = s/(2√3) = s√3/6. The inradius is used in problems where the incircle is given or where the area must be computed from the perimeter and inradius.

66. Inscribed Angle

An angle whose vertex lies on a circle and whose sides are chords of the circle. The inscribed angle theorem states that the inscribed angle equals half the central angle that subtends the same arc. Equivalently, the inscribed angle equals half the intercepted arc in degrees. Inscribed angles subtending the same arc are equal and this corollary is one of the most used circle results on the board exam.

67. Inscribed Angle Theorem — Corollaries

Key corollaries of the inscribed angle theorem: (1) inscribed angles subtending the same arc are equal, (2) an inscribed angle subtending a semicircle is 90°, (3) opposite angles of a cyclic quadrilateral are supplementary. These three corollaries are more frequently tested than the base theorem itself and should be recognized instantly in circle problems.

68. Inscribed Circle (Incircle)

A circle tangent to all three sides of a triangle from the interior. The center is the incenter and the radius is r = A/s, where A is the triangle area and s is the semi-perimeter. For an equilateral triangle with side s, r = s/(2√3). The incircle is the largest circle that fits inside the triangle and is used in problems involving tangent lengths from vertices.

69. Inscribed Polygon

A polygon whose vertices all lie on a circle. A triangle is always inscribed in its circumcircle. A quadrilateral is inscribed in a circle only if opposite angles are supplementary. Inscribed regular polygons divide the circle into equal arcs and are used in area comparison problems and in problems involving the relationship between polygon side length and circle radius.

70. Intercept Theorem (Basic Proportionality Theorem)

If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio. If DE is parallel to BC with D on AB and E on AC, then AD/DB = AE/EC. The intercept theorem is the foundation of similar triangle theory and is used in problems involving parallel lines cutting two transversals proportionally.

151. Viviani’s Theorem

For any point inside an equilateral triangle, the sum of the perpendicular distances from the point to the three sides equals the altitude of the triangle. This elegant result means the sum of the three distances is constant regardless of where the interior point is located. It appears in competition-style board problems involving equilateral triangles and uniformly distributed interior measurements.

CONCLUSION

Plane geometry has a small core of ideas that generate a very large number of testable results. The most important cluster for the PRC board exam is the circle theorems; the inscribed angle theorem and its three corollaries, the chord-chord angle theorem, the tangent-chord angle theorem, the secant-secant and tangent-secant angle theorems, and the three versions of the power of a point. These theorems generate the majority of circle problems on the exam and can all be derived from the single idea that inscribed angles equal half their intercepted arc. Mastering this cluster alone will significantly raise your score on geometry problems.

The second cluster to prioritize is the triangle center properties and the classical triangle theorems. Know the four centers namely centroid, circumcenter, incenter, orthocenter and their definitions, their formulas, and their locations for acute, right, and obtuse triangles. Know the Euler line and the nine-point circle as connecting results. For the classical theorems, Heron’s formula, Stewart’s theorem, Ceva’s theorem, Menelaus’ theorem, and the median length formula each generate a specific class of board problem that cannot be solved quickly without them. These theorems appear in problems that initially look like they need trigonometry or algebra but yield immediately to the right geometric tool.

The third cluster is polygon properties such as interior angle sums, diagonal count formulas, regular polygon area and apothem formulas, properties of cyclic quadrilaterals including Brahmagupta’s formula and Ptolemy’s theorem, and properties of the special quadrilaterals. Locus problems form a fourth standalone cluster and are reliably tested by asking you to identify the curve or line satisfying a geometric condition. The six standard locus types in this list cover all the configurations the board exam uses. Review each cluster systematically, connect every term to at least one worked example, and the geometry portion of the board exam will become one of your most consistent scoring areas.

For practice problems on all these topics, head over to our Plane Geometry Problems and Solutions section here on PinoyBix. Hundreds of solved exam-type questions, complete with step-by-step solutions, organized by topic so you can drill exactly what you need to work on.

Everything else in this list supports those core topics. If you are solid on the fundamentals, you will be in a strong position to handle whatever plane geometry problems appear on exam day. Good luck, and keep at it.