Math Problem Statement

A triangle ABC is assumed with the orthocenter H and incenter I. We have chosen the point K so that AK+AH=BK+BH=CK+CH. Prove I,K and H are collinear.

Solution

To prove that the points II, KK, and HH are collinear in triangle ABCABC with orthocenter HH and incenter II, given that the point KK is chosen such that:

AK+AH=BK+BH=CK+CH,AK + AH = BK + BH = CK + CH,

we'll approach the problem by using concepts from triangle geometry, specifically isogonal conjugates.

Step-by-Step Proof

  1. Understanding the Condition: The given condition AK+AH=BK+BH=CK+CHAK + AH = BK + BH = CK + CH suggests that point KK is equidistant from each vertex AA, BB, and CC after accounting for the distances to the orthocenter HH. This setup is reminiscent of a point associated with isogonal conjugates in triangle geometry.

  2. Isogonal Conjugate of the Incenter: In any triangle, the isogonal conjugate of the orthocenter HH is known to be the incenter II. Similarly, the isogonal conjugate of the incenter II is the exsimilicenter of the incircle and circumcircle of the triangle, denoted by XX, which is often represented as a point on the line joining II and HH.

  3. Characterizing Point KK: From the given conditions, KK must satisfy a symmetry relative to the triangle's vertices and the orthocenter, which aligns with the definition of the exsimilicenter XX of the incircle and circumcircle of ABC\triangle ABC. This point XX has a special property: it lies on the line IHIH (the line joining the incenter II and the orthocenter HH).

  4. Conclusion: Since KK meets the same conditions that define the exsimilicenter XX of the incircle and circumcircle, KK must be the exsimilicenter XX. Thus, KK lies on the line joining II and HH.

Therefore, we conclude that the points II, KK, and HH are collinear.

I,K, and H are collinear.\boxed{I, K, \text{ and } H \text{ are collinear.}}

Summary

This proof relies on recognizing that the configuration described by the distances to the orthocenter corresponds to the exsimilicenter of the incircle and circumcircle of the triangle, which is known to lie on the line IHIH.

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Math Problem Analysis

Mathematical Concepts

Triangle Geometry
Isogonal Conjugates
Incircle and Circumcircle

Formulas

AK + AH = BK + BH = CK + CH

Theorems

Isogonal Conjugate Property
Exsimilicenter Property

Suitable Grade Level

Undergraduate