Concept 10 of 10Foundation

📌 Formula Sheet — Constructions

Key rules and properties of the constructions you use — with the why behind each.

Triangle inequality: a + b > c for every pair of sides

Means: the sum of any two sides must exceed the thirdUse when: checking if a triangle with given sides is possible

Why: A straight line is the shortest path between two points. Going via a third point (the other sides) must be LONGER than going direct (the third side).

⚠ Trap: If a + b = c exactly, it's degenerate (flat line, not a triangle). Must be STRICTLY greater.

Perpendicular bisector = locus of points equidistant from both endpoints

Means: any point on it is equally far from A and BUse when: finding circumcentre, equal-distance problems

Why: By construction, we drew equal-radius arcs from A and B. Their intersection is thus equidistant. Extend through the midpoint and you get a line of such points.

⚠ Trap: The line is PERPENDICULAR to the segment AND passes through the midpoint. Both conditions are needed.

Angle bisector = locus of points equidistant from both arms

Means: any point on it has equal perpendicular distance to the two sides of the angleUse when: finding incentre, fair placement problems

Why: Symmetry: the bisector mirrors one arm onto the other. A point on it sees both arms equally.

⚠ Trap: Bisector divides the ANGLE in half — not necessarily the opposite side of a triangle equally.

Triangle centres at-a-glance

Incentre: meeting of angle bisectors — centre of inscribed circleCircumcentre: meeting of perpendicular bisectors — centre of circle through all 3 verticesCentroid: meeting of medians — centre of mass (divides each median 2:1 from vertex)Orthocentre: meeting of altitudes

Why: Each "centre" is the intersection of three lines that all share a common property (e.g., all 3 angle bisectors are equidistant from sides → incentre).

⚠ Trap: In a RIGHT triangle, the circumcentre is the midpoint of the hypotenuse; the orthocentre is the right-angle vertex.

Circle basics: Diameter = 2 × Radius

Means: d = 2rUse when: switching between radius and diameter

Why: Diameter passes through centre end to end — two radii in a straight line.

⚠ Trap: Chord ≤ diameter. Diameter is the LONGEST chord.

Minimum measurements to fix a triangle: 3 parts (SSS, SAS, ASA, or RHS)

Means: Three correct measurements determine a unique triangleUse when: checking whether a triangle can be constructed from given info

Why: Less than 3 leaves too many possibilities. Exactly 3 (with correct type) locks every side and angle uniquely.

⚠ Trap: SSA (two sides + non-included angle) can give 0, 1, or 2 triangles — not always unique ("ambiguous case").

💡 Tip:Every geometric centre of a triangle coincides if the triangle is equilateral.

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