Unique Geometry Problem.

Geometry Level pending

Two tangents are drawn to a circle from an exterior point K; they touch the circle at the points D and E. A third tangent intersects segment KD at X, and segment KE at Y. It touches the circle at Z. If KD = 10, find the perimeter of triangle KXY.


The answer is 20.

Asher Joy by Asher Joy , 22, USA

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3 solutions

Michael Mendrin
May 16, 2014

Since the problem implies that the perimeter is invariant, we can shrink the circle to a point, so that the perimeter is 10 + 0 + 10 = 20. Proving it to be invariant is another matter, but that's not the problem given.

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Or we can roll the tangent XY on the circle to its limiting position where X coincides with D and Y coincides with K (Giving us a degenerate triangle which is line KD doubled back on itself)

This will give us the perimeter = 2 KD = 20

Ujjwal Rane - 6 years, 6 months ago
Ujjwal Rane
Dec 10, 2014

Since two tangents to a given circle, drawn from a point are equal in length:

XD = XZ; YZ = YE; KD = KE

Hence KX + XZ = KX + XD = KD - - - (I)

and KY + YZ = KY + YE = KE - - - (II)

Adding (I) + (II) we get the perimeter

Perimeter K-X-Z-Y-K = KD + KE = 20

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Tushar Pandey
May 27, 2014

KX+XY+KY= KD-DX + DX+EY + KE-EY => KX+XY+KY=KD+KE=10+10=20 Given: KD=KE DX=XZ EY=YZ

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