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Explore the Back-Story1 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.

Discussion:

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 .

Hence triangle PQR is equilateral.

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.

Discussion:

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 .

Hence triangle PQR is equilateral.

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the solution to first question isn't complete...you will also hav to contradict the case where tangents are opposite