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The external bisector of angle A of A triangle ABC meets BC produced at L. And the internal bisector of angle B meets CA at M. If LM meets AB at R. Prove that CR bisects angle C?
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The external bisector of angle A of A triangle ABC meets BC produced a...
Given:
To prove: CR bisects angle C
Proof:
- Triangle ABC
- External bisector of angle A meets BC produced at L
- Internal bisector of angle B meets CA at M
- LM meets AB at R
To prove: CR bisects angle C
Proof:
- Lemma 1: The external bisector of an angle of a triangle divides the opposite side externally in the ratio of the adjacent sides.
- Lemma 2: The internal bisectors of two angles of a triangle intersect at a point which is equidistant from the sides of the third angle.
- Proof of the main result: Since CR is the third transversal of the lines AL and BM, we have:
Proof of Lemma 1: Let ABC be a triangle and AD be its external bisector of angle A. Then, by angle bisector theorem, we have:
BD/DC = AB/AC
By exterior angle theorem, we have:
∠BAC = ∠ABD + ∠ADC
But, ∠ABD = ∠ADC (as AD is an external bisector)
So, ∠BAC = 2∠ABD
Therefore, in triangle ABD, we have:
sin(∠ABD)/BD = sin(∠BAC/2)/AB
Similarly, in triangle ADC, we have:
sin(∠ADC)/DC = sin(∠BAC/2)/AC
Dividing the two equations, we get:
BD/DC = AB/AC
which proves the lemma.
Proof of Lemma 2: Let ABC be a triangle and AD and BE be its internal bisectors of angles A and B respectively. Then, by angle bisector theorem, we have:
BD/DC = AB/AC ...(1)
and
AE/EC = AB/BC ...(2)
Multiplying (1) and (2), we get:
(BD/DC)*(AE/EC) = 1
By converse of angle bisector theorem, we have:
AD and BE intersect at a point I which is equidistant from sides AC and BC.
(BL/LC) * (CA/AM) * (MR/RB) = 1
By Lemma 1, we have:
BL/LC = AB/AC
By Lemma 2, we have:
CA/AM = CB/BM
Therefore, we have:
(AB/AC) * (CB/BM) * (MR/RB) = 1
which implies:
MR/RB = AC/AB
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The external bisector of angle A of A triangle ABC meets BC produced at L. And the internal bisector of angle B meets CA at M. If LM meets AB at R. Prove that CR bisects angle C?
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The external bisector of angle A of A triangle ABC meets BC produced at L. And the internal bisector of angle B meets CA at M. If LM meets AB at R. Prove that CR bisects angle C? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about The external bisector of angle A of A triangle ABC meets BC produced at L. And the internal bisector of angle B meets CA at M. If LM meets AB at R. Prove that CR bisects angle C? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The external bisector of angle A of A triangle ABC meets BC produced at L. And the internal bisector of angle B meets CA at M. If LM meets AB at R. Prove that CR bisects angle C?.
The external bisector of angle A of A triangle ABC meets BC produced at L. And the internal bisector of angle B meets CA at M. If LM meets AB at R. Prove that CR bisects angle C? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about The external bisector of angle A of A triangle ABC meets BC produced at L. And the internal bisector of angle B meets CA at M. If LM meets AB at R. Prove that CR bisects angle C? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The external bisector of angle A of A triangle ABC meets BC produced at L. And the internal bisector of angle B meets CA at M. If LM meets AB at R. Prove that CR bisects angle C?.
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