Let triangle ABC have 1 as the center of the inscribed circle, O as the center of the circumcircle. (1) contact BC, CA, AB at D, E, F, H is the projection of D onto EF. AK is the diameter of (0) DS is the diameter of (1). Prove that: 1. Of is the Euler line of triangle DEF 2. H, I, K are aligned. 3. AS passes through the point of contact of the circle at angle A with BC 4. DH is the bisector of BHC. 5. Bi contains EF at X, then XB is perpendicular to XC and X lies on the median line corresponding to vertex C of triangle ABC 6. K is the orthocenter of triangle ABC, then DH is the bisector of IHK 7. Dietersects Ef at T, then AT passes through the midpoint of BC 8. Mi intersects AH at Q, then AQ-ID (AH is the attitude, H is the foot of the altitude). 9. (Al) intersects (0) at P then PD passes through the midpoint of minor aro BC, 10. H belongs to the square axis of (ABE), (ACF) 11. H belongs to the square axis of (BE), (CF) 12. Y is the projection of Manto EF, then IY passes through the middle of the major arc BC. 13. D'in symmetrical with D through EF, then AD intersects BC on Di 14. Bland Ci intersect AC and AB at E and F respectively. D is the intersection of the tangent at B and C of (0). Prove ID bisects EF
Let triangle ABC have 1 as the center of the inscribed circle, O as the center of the circumcircle. (1) contact BC, CA, AB at D, E, F, H is the projection of D onto EF. AK is the diameter of (0) DS is the diameter of (1). Prove that: 1. Of is the Euler line of triangle DEF 2. H, I, K are aligned. 3. AS passes through the point of contact of the circle at angle A with BC 4. DH is the bisector of BHC. 5. Bi contains EF at X, then XB is perpendicular to XC and X lies on the median line corresponding to vertex C of triangle ABC 6. K is the orthocenter of triangle ABC, then DH is the bisector of IHK 7. Dietersects Ef at T, then AT passes through the midpoint of BC 8. Mi intersects AH at Q, then AQ-ID (AH is the attitude, H is the foot of the altitude). 9. (Al) intersects (0) at P then PD passes through the midpoint of minor aro BC, 10. H belongs to the square axis of (ABE), (ACF) 11. H belongs to the square axis of (BE), (CF) 12. Y is the projection of Manto EF, then IY passes through the middle of the major arc BC. 13. D'in symmetrical with D through EF, then AD intersects BC on Di 14. Bland Ci intersect AC and AB at E and F respectively. D is the intersection of the tangent at B and C of (0). Prove ID bisects EF