MathDB

3

Part of 2001 Pan African

Problems(2)

Grasshopper jumps in an equilateral triangle

Source: Pan African 2001

10/4/2005
Let ABCABC be an equilateral triangle and let P0P_0 be a point outside this triangle, such that △AP0C\triangle{AP_0C} is an isoscele triangle with a right angle at P0P_0. A grasshopper starts from P0P_0 and turns around the triangle as follows. From P0P_0 the grasshopper jumps to P1P_1, which is the symmetric point of P0P_0 with respect to AA. From P1P_1, the grasshopper jumps to P2P_2, which is the symmetric point of P1P_1 with respect to BB. Then the grasshopper jumps to P3P_3 which is the symmetric point of P2P_2 with respect to CC, and so on. Compare the distance P0P1P_0P_1 and P0PnP_0P_n. n∈Nn \in N.
geometrygeometric transformation
A lot of circle

Source: Pan African 2001

10/4/2005
Let S1S_1 be a semicircle with centre OO and diameter ABAB.A circle C1C_1 with centre PP is drawn, tangent to S1S_1, and tangent to ABAB at OO. A semicircle S2S_2 is drawn, with centre QQ on ABAB, tangent to S1S_1 and to C1C_1. A circle C2C_2 with centre RR is drawn, internally tangent to S1S_1 and externally tangent to S2S_2 and C1C_1. Prove that OPRQOPRQ is a rectangle.
quadraticsgeometryperpendicular bisectoralgebra