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

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

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 = AK / AC
KP / BC = 18 / (1,2BC)
KP = 18BC / (1,2BC) = 15
Answer: KP = 15