Maths Olympiad Exam

Let AC be a line segment in the plane and B a point between A and C. Construct isosceles triangles PAB and QBC on one side of the segment AC such that APB = BQC = 120 and an isosceles triangle RAC on the otherside of AC such that ARC = 120. Show that PQR is an equilateral triangle.

Read Solution (Total 0)

Maths Olympiad Other Question

HW:8'VXHM:701N>fBiHI4~70jm:²lp+VY Produce AP and CQ to meet at K. Observe that AKCR is a rhombus and BQKP is a parallelogram. Put AP = x,CQ = y. Then PK = BQ = y, KQ = PB = x and AR = RC = CK = KA = x + y. Using cosine rule in triangle PKQ, we get PQ^2 = x^2 + y^2 - 2xy cos120 = x^2 + y^2 + xy. Similarly cosine rule in triangle QCR gives QR^2 = y^2 +(x+y)^2 - 2xy cos60 = x^2 +y^2+xy and cosine rule in triangle PAR gives RP^2 = x^2 + (x + y)^2 - 2xy cos 60 = x^2 + y^2 + xy. It follows that PQ = QR = RP.