MathDB

4

Part of 1999 Austrian-Polish Competition

Problems(1)

find a point such thath 3 triangles have the same area

Source: Austrian-Polish 1999

5/24/2019
Three lines k,l,mk, l, m are drawn through a point PP inside a triangle ABCABC such that kk meets ABAB at A1A_1 and ACAC at A2≠A1A_2 \ne A_1 and PA1=PA2PA_1 = PA_2, ll meets BCBC at B1B_1 and BABA at B2≠B1B_2 \ne B_1 and PB1=PB2PB_1 = PB_2, mm meets CACA at C1C_1 and CBCB at C2≠C1C_2\ne C_1 and PC1=PC2PC_1=PC_2. Prove that the lines k,l,mk,l,m are uniquely determined by these conditions. Find point PP for which the triangles AA1A2,BB1B2,CC1C2AA_1A_2, BB_1B_2, CC_1C_2 have the same area and show that this point is unique.
geometryarea of a triangleareasconstruction