# Difference between revisions of "1985 IMO Problems/Problem 5"

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<math>M</math> is the Miquel Point of quadrilateral <math>ACNK</math>, so there is a spiral similarity centered at <math>M</math> that takes <math>KA</math> to <math>NC</math>. Let <math>M_1</math> be the midpoint of <math>KA</math> and <math>M_2</math> be the midpoint of <math>NC</math>. Thus the spiral similarity must also send <math>M_1</math> to <math>M_2</math> and must also be the center of another spiral similarity that sends <math>KN</math> to <math>M_1 M_2</math>, so <math>BMM_1 M_2</math> is cyclic. <math>OM_1 B M_2</math> is also cyclic with diameter <math>BO</math> and thus <math>M</math> must lie on the same circumcircle as <math>B</math>, <math>M_1</math>, and <math>M_2</math> so <math>\angle OMB = 90^{\circ}</math>. | <math>M</math> is the Miquel Point of quadrilateral <math>ACNK</math>, so there is a spiral similarity centered at <math>M</math> that takes <math>KA</math> to <math>NC</math>. Let <math>M_1</math> be the midpoint of <math>KA</math> and <math>M_2</math> be the midpoint of <math>NC</math>. Thus the spiral similarity must also send <math>M_1</math> to <math>M_2</math> and must also be the center of another spiral similarity that sends <math>KN</math> to <math>M_1 M_2</math>, so <math>BMM_1 M_2</math> is cyclic. <math>OM_1 B M_2</math> is also cyclic with diameter <math>BO</math> and thus <math>M</math> must lie on the same circumcircle as <math>B</math>, <math>M_1</math>, and <math>M_2</math> so <math>\angle OMB = 90^{\circ}</math>. |

## Revision as of 21:38, 26 February 2013

## Problem

A circle with center passes through the vertices and of the triangle and intersects the segments and again at distinct points and respectively. Let be the point of intersection of the circumcircles of triangles and (apart from ). Prove that .

## Solution

is the Miquel Point of quadrilateral , so there is a spiral similarity centered at that takes to . Let be the midpoint of and be the midpoint of . Thus the spiral similarity must also send to and must also be the center of another spiral similarity that sends to , so is cyclic. is also cyclic with diameter and thus must lie on the same circumcircle as , , and so .