Question

Let two circles intersect each other at point a and d . let the diameter ab intersect the circle with center p in point n and diameter ac intersect the circle with center o in the point m then prove that ac× am = ab × an.

Answer 1

Given -

two circles intersect each other at point a and d

Two circles have centres as p and o

To prove-

ac × am= ab ×an

Proof-

We get two triangles

ΔACN and ΔABM

We know that semicircle angle is a right angle

Therefore ∠AMB = ∠ANC

also, ∠MAB =∠NAC(vertically opposite angles)

Therefore ΔACN and ΔABM are similar triangles by angle- angle similarity property

As both triangles are similar we get

AC/AB =AN/AM

Therefore ac ×am = ab ×an