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Theorem. If AABC is an acute triangle, then the altitudes of AABC are the angle bisectors of the orthic triangle AA'B'C'. Proof. Consider an acute triangle AABC. Let A', B', C' be the feet of the altitudes through A, B, C, respectively. Then we have orthic triangle AA'B'C'. Insert image of the setup Write an argument for why the altitude through C bisects angle ZA'C'B'. Insert any supporting pictures for your argument. Similarly, we can show the altitude through A bisects angle ZB'A'C", and the altitude through B bisects angle ZA'B'C'. Thus the altitudes of triangle AABC are the angle bisectors of triangle AA'B'C". Corollary. If AABC is an acute triangle, then the insert appropriate tri- angle center of triangle AABC is the insert appropriate triangle center of triangle AA'B'C".

Theorem. If AABC is an acute triangle, then the altitudes of AABC are the angle bisectors of the orthic triangle AA'B'C'. Proof. Consider an acute triangle AABC. Let A', B', C' be the feet of the altitudes through A, B, C, respectively. Then we have orthic triangle AA'B'C'. Insert image of the setup Write an argument for why the altitude through C bisects angle ZA'C'B'. Insert any supporting pictures for your argument. Similarly, we can show the altitude through A bisects angle ZB'A'C", and the altitude through B bisects angle ZA'B'C'. Thus the altitudes of triangle AABC are the angle bisectors of triangle AA'B'C". Corollary. If AABC is an acute triangle, then the insert appropriate tri- angle center of triangle AABC is the insert appropriate triangle center of triangle AA'B'C".

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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used geogebra for the images and file out the blueusing euclid geometry 

**Theorem:** If \( \triangle ABC \) is an acute triangle, then the altitudes of \( \triangle ABC \) are the angle bisectors of the orthic triangle \( \triangle A'B'C' \). **Proof:** Consider an acute triangle \( \triangle ABC \). Let \( A', B', C' \) be the feet of the altitudes through \( A, B, C \), respectively. Then we have orthic triangle \( \triangle A'B'C' \). *Insert image of the setup.* Write an argument for why the altitude through \( C \) bisects angle \( \angle A'C'B' \). *Insert any supporting pictures for your argument.* Similarly, we can show the altitude through \( A \) bisects angle \( \angle B'A'C' \), and the altitude through \( B \) bisects angle \( \angle A'B'C' \). Thus, the altitudes of triangle \( \triangle ABC \) are the angle bisectors of triangle \( \triangle A'B'C' \). \(\square\) **Corollary:** If \( \triangle ABC \) is an acute triangle, then the *insert appropriate triangle center* of triangle \( \triangle ABC \) is the *insert appropriate triangle center* of triangle \( \triangle A'B'C' \).
Transcribed Image Text:**Theorem:** If \( \triangle ABC \) is an acute triangle, then the altitudes of \( \triangle ABC \) are the angle bisectors of the orthic triangle \( \triangle A'B'C' \). **Proof:** Consider an acute triangle \( \triangle ABC \). Let \( A', B', C' \) be the feet of the altitudes through \( A, B, C \), respectively. Then we have orthic triangle \( \triangle A'B'C' \). *Insert image of the setup.* Write an argument for why the altitude through \( C \) bisects angle \( \angle A'C'B' \). *Insert any supporting pictures for your argument.* Similarly, we can show the altitude through \( A \) bisects angle \( \angle B'A'C' \), and the altitude through \( B \) bisects angle \( \angle A'B'C' \). Thus, the altitudes of triangle \( \triangle ABC \) are the angle bisectors of triangle \( \triangle A'B'C' \). \(\square\) **Corollary:** If \( \triangle ABC \) is an acute triangle, then the *insert appropriate triangle center* of triangle \( \triangle ABC \) is the *insert appropriate triangle center* of triangle \( \triangle A'B'C' \).
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