# The circle intersects sides AB and AC of triangle ABC at points K and P, respectively, and passes through vertices B and C

**The circle intersects sides AB and AC of triangle ABC at points K and P, respectively, and passes through vertices B and C. Find the length of the segment KP if AP = 18 and side BC is 1.2 times smaller than side AB**

Consider the quadrilateral PKBC.

PKBC is inscribed in a circle, therefore the condition is fulfilled: the sum of the opposite corners of the quadrangle is 180 ° (the condition that the quadrangle can be inscribed in a circle).

Those. ∠PKB + ∠BCP = 180 °

∠PKB + ∠AKP = 180 ° (as these are adjacent angles).

Therefore, ∠AKP = ∠BCP

Consider the triangles ABC and AKP.

∠AKP = ∠BCP (we found out a little higher)

∠A is common, then these triangles are similar (on the basis of similarity).

Therefore, KP / BC = AK / AC = AP / AB (from the definition of such triangles).

We are interested in the equality KP / BC = AP / AB

KP / BC = 18 / (1,2BC)

KP = 18BC / (1,2BC) = 15

Answer: KP = 15