1 Let and be two circles touching each other externally at R. Let and be the centres of and , respectively. Let be a line which is tangent to at P and passing through , and let be the line tangent to at Q and passing through . Let . If KP=KQ then prove that the triangle PQR is equilateral.
We note that is a straight line (why?)
Also are right angled triangles with right angles at point Q and P respectively.
Hence are similar (vertically opposite angles and right angles)
Thus as KP =KQ.
Hence the radii of the two circles are equal..
This implies R is the midpoint of hence the midpoint of hypotenuse of
since all are circum-radii of .
Hence is equilateral, similarly is also equilateral.
Thus is also \( RQ = O_1 R = O_2 R = RP \).
Hence triangle PQR is equilateral.