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 , , and are collinear in triangle with orthocenter and incenter , given that the point is chosen such that:
we'll approach the problem by using concepts from triangle geometry, specifically isogonal conjugates.
Step-by-Step Proof
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Understanding the Condition: The given condition suggests that point is equidistant from each vertex , , and after accounting for the distances to the orthocenter . This setup is reminiscent of a point associated with isogonal conjugates in triangle geometry.
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Isogonal Conjugate of the Incenter: In any triangle, the isogonal conjugate of the orthocenter is known to be the incenter . Similarly, the isogonal conjugate of the incenter is the exsimilicenter of the incircle and circumcircle of the triangle, denoted by , which is often represented as a point on the line joining and .
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Characterizing Point : From the given conditions, must satisfy a symmetry relative to the triangle's vertices and the orthocenter, which aligns with the definition of the exsimilicenter of the incircle and circumcircle of . This point has a special property: it lies on the line (the line joining the incenter and the orthocenter ).
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Conclusion: Since meets the same conditions that define the exsimilicenter of the incircle and circumcircle, must be the exsimilicenter . Thus, lies on the line joining and .
Therefore, we conclude that the points , , and 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 .
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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