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Let ABC have incircle C(0,r), and let the points of tangency of C(0,r) with BC, AC, AB be D, E and F respectively. Use the following steps to give a direct proof that DEF must be an acute triangle. (a) Label ZEOF as 21 in your diagram. Prove that 21 = 180°-ZA. Similarly, what are ZDOF and ZDOE in terms of ZB and ZC? (b) Express ZEDF in terms of 21. Using part (a). what is ZEDF in terms of ZA? Similarly, what are ZDEF and ZDFE in terms of ZB and ZC respectively? (c) Using part (b) conclude that ZEDF < 90°, ZDEF < 90° and ZDFE < 90°, and hence ADEF is an acute triangle.

Let ABC have incircle C(0,r), and let the points of tangency of C(0,r) with BC, AC, AB be D, E and F respectively. Use the following steps to give a direct proof that DEF must be an acute triangle. (a) Label ZEOF as 21 in your diagram. Prove that 21 = 180°-ZA. Similarly, what are ZDOF and ZDOE in terms of ZB and ZC? (b) Express ZEDF in terms of 21. Using part (a). what is ZEDF in terms of ZA? Similarly, what are ZDEF and ZDFE in terms of ZB and ZC respectively? (c) Using part (b) conclude that ZEDF < 90°, ZDEF < 90° and ZDFE < 90°, and hence ADEF is an acute triangle.

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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Let AABC have incircle C(O, r), and let the points of tangency of C(O, r) with BC, AC, AB be D, E and F respectively. Use the following steps to give a direct proof that DEF must be an acute triangle. (a) Label ZEOF as 21 in your diagram. Prove that <1 = 180°-ZA. Similarly, what are ZDOF and ZDOE in terms of ZB and ZC? (b) Express ZEDF in terms of 21. Using part (a). what is ZEDF in terms of ZA? Similarly, what are ZDEF and ZDFE in terms of ZB and ZC respectively? (c) Using part (b) conclude that ZEDF < 90°, ZDEF < 90° and ZDFE < 90°, and hence ADEF is an acute triangle.
Transcribed Image Text:Let AABC have incircle C(O, r), and let the points of tangency of C(O, r) with BC, AC, AB be D, E and F respectively. Use the following steps to give a direct proof that DEF must be an acute triangle. (a) Label ZEOF as 21 in your diagram. Prove that <1 = 180°-ZA. Similarly, what are ZDOF and ZDOE in terms of ZB and ZC? (b) Express ZEDF in terms of 21. Using part (a). what is ZEDF in terms of ZA? Similarly, what are ZDEF and ZDFE in terms of ZB and ZC respectively? (c) Using part (b) conclude that ZEDF < 90°, ZDEF < 90° and ZDFE < 90°, and hence ADEF is an acute triangle.
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