Triangles & Circles
Congruence, similarity, angle in a circle, tangents, and power of a point
Key Concepts
- Congruence and similarity criteria for triangles
- Circle theorems: angles in the same segment, cyclic quadrilaterals
- Power of a point relates secants and tangents
- Standard centres: centroid, incentre, circumcentre, orthocentre
Important Formulae
| Area of a triangle | ½ · base · height = ½ ab sinC |
| Inradius / circumradius | r = Area/s; R = abc/(4·Area) |
| Ptolemy (cyclic quad) | AC · BD = AB · CD + AD · BC |
Quick Tips
- A tangent is always perpendicular to the radius at the point of contact.
- Start most problems with angle chasing and look for cyclic quadrilaterals.
Sample Practice Questions
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A chord AB subtends angle 60° at the circumference. The angle at the centre is:
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Answer: 120°
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The sine rule: a/sin A = b/sin B = c/sin C = ?
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Answer: 2R (diameter of circumcircle)
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Ceva's theorem: for cevians AD, BE, CF concurrent in triangle ABC:
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Answer: (AF/FB) × (BD/DC) × (CE/EA) = 1
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The product of the diagonals of a cyclic quadrilateral equals:
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Answer: Sum of products of opposite sides (Ptolemy)
Practice Questions
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What are the key concepts in Triangles & Circles for PRMO?
Congruence and similarity criteria for triangles; Circle theorems: angles in the same segment, cyclic quadrilaterals; Power of a point relates secants and tangents; Standard centres: centroid, incentre, circumcentre, orthocentre.
Which formulae are important for Triangles & Circles in PRMO?
Area of a triangle: ½ · base · height = ½ ab sinC; Inradius / circumradius: r = Area/s; R = abc/(4·Area); Ptolemy (cyclic quad): AC · BD = AB · CD + AD · BC.
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