# 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 Remember: The process of learning a person lasts a lifetime. The value of the same knowledge for different people may be different, it is determined by their individual characteristics and needs. Therefore, knowledge is always needed at any age and position.