Geometry: Similarity, Circles, and Constructions
Similarity and right triangles
Two triangles are similar when corresponding angles are equal and corresponding sides are proportional; write vertices in matching order. Thales’ theorem (basic proportionality): a line parallel to one side of a triangle divides the other two sides proportionally. Its converse: if a line divides two sides proportionally, it is parallel to the third side. Pythagoras: in a right triangle, hypotenuse²=sum of squares of the other two sides. Converse: if the square of one side equals the sum of squares of the other two, the triangle is right-angled.
Circles and tangents
A tangent touches a circle at one point and is perpendicular to the radius at that point. Tangents from an external point are equal. The tangent–chord theorem says the angle between a tangent and a chord equals the angle in the alternate segment. The perpendicular bisectors of a triangle are concurrent at the circumcentre; angle bisectors at the incentre; medians at the centroid; altitudes at the orthocentre. Constructions often ask for a tangent from an external point, a similar triangle, or a triangle with given data; use ruler and compass and leave construction arcs.
Worked example
From an external point P, a tangent PT is drawn to a circle of radius 5 cm with OP=13 cm. Since OT⊥PT, triangle OTP is right-angled at T. PT²=OP²−OT²=169−25=144, so PT=12 cm.
Traps
Matching wrong vertices in similarity; using the hypotenuse as a leg; forgetting the tangent is perpendicular to the radius; and in constructions, erasing the arcs that prove the construction.