Example 5
Prove that the quadrilateral formed (if possible) by the internal angle bisectors of any quadrilateral is cyclic.
Given: ABCD is a quadrilateral
AH, BF, CF, DH are bisectors of
A , B, C, D respectively
To prove: EFGH is cyclic quadrilateral
Proof: To prove EFGH is a cyclic quadrilateral,
we prove that sum of one pair of opposite angles is 180
In AEB
ABE + BAE + AEB = 180
AEB = 180 ABE BAE
AEB = 180 (1/2 B + 1/2 A)
AEB = 180 1/2 ( B + A)
Now,
lines AH & BF intersect
So, FEH = AEB
FEH = 180 1/2 ( B + A)
Similarly, we can prove that
FGH = 180 1/2 ( C + D)
Adding (2) & (3)
FEH + FGH = 180 1/2 ( A + D) + 180 1/2 ( C + B)
FEH + FGH = 180 + 180 1/2 ( A + D + C + B )
FEH + FGH = 360 1/2 ( A + B + C + D )
FEH + FGH = 360 1/2 ( A + B + C + D )
FEH + FGH = 360 1/2 360
FEH + FGH = 360 180
FEH + FGH = 180
Thus, in EFGH,
Since sum of one pair of opposite angles is 180
EFGH is a cyclic quadrilateral

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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