Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN: 9781337614085
Author: Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher: Cengage,
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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 \( \triangle A'B'C' \).
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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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Step 1: The diagram of the given situation drawn using Geogebra.
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Step 2: Argument (Taken example of angle C'B'A'
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Step 3: Steps to draw an altitude from point C on AB
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