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Prove that r1+r2+r3-r=4R...

Prove that `r_1+r_2+r_3-r=4R`

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To prove that \( r_1 + r_2 + r_3 - r = 4R \), we will follow a systematic approach using the relationships between the radii of the incircle and exradii of a triangle. ### Step-by-step Solution: 1. **Understanding the Terms**: - Let \( r \) be the radius of the incircle of triangle \( ABC \). - Let \( r_1, r_2, r_3 \) be the exradii opposite to vertices \( A, B, C \) respectively. - The semi-perimeter \( s \) of triangle \( ABC \) is given by \( s = \frac{a + b + c}{2} \), where \( a, b, c \) are the lengths of the sides opposite to vertices \( A, B, C \). ...
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CENGAGE ENGLISH-PROPERTIES AND SOLUTIONS OF TRIANGLE-Illustration
  1. Prove that the distance between the circumcenter and the incenter of ...

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  2. Prove that acosA+bcosB+ccosClt=sdot

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  3. If Delta is the area of a triangle with side lengths a, b, c, then s...

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  4. If in Delta ABC, the distance of the vertices from the orthocenter are...

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  5. ABC is an acute angled triangle with circumcenter O and orthocentre H....

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  6. In a acute angled triangle ABC, proint D, E and F are the feet of the ...

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  7. Prove that the distance between the circumcenter and the orthocentre o...

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  8. Let ABC be an acute angled triangle whose orthocentre is at H. If a...

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  9. In A B C , let L ,M ,N be the feet of the altitudes. The prove that s...

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  10. The lengths of the medians through acute angles of a right-angled tr...

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  11. Two medians drawn from acute angles of a right angled triangles inters...

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  12. Prove that r1+r2+r3-r=4R

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  13. If in a triangle r1=r2+r3+r , prove that the triangle is right angled.

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  14. Prove that (r(1+r2))/1=2R

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  15. Prove that (r+r1)tan((B-C)/2)+(r+r2)tan((C-A)/2)+(r+r3)tan((A-B)/2)=0

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  16. If the distance between incenter and one of the excenter of an equi...

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  17. If I(1), I(2), I(3) are the centers of escribed circles of Delta ABC, ...

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  18. Prove that the sum of the radii of the radii of the circles, which are...

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  19. If the area of the circle is A1 and the area of the regular pentagon i...

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  20. Prove that the area of a regular polygon hawing 2n sides, inscribed in...

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