Question

In a quadrilateral ABCD if the bisector of <ABC and <ADC meet on diagonal AC ,prove that the bisector of the <BAD and <BCD will meet in diagonal BD.

Answer 1

[fig.is in the attachment]

Given: ABCD is a quadrilateral in which the bisectors of ∠ABC & ∠ADC meet on the diagonal AC at P.

To prove: bisectors of ∠BAD & ∠BCD meet on the diagonal BD.

Construction: join BP and DP. Let the bisectors of ∠BAD meet BD at Q. Join AQ & CQ

Proof:
In order to prove that the bisectors of ∠BAD & ∠BCD meet on the diagonal BD.
Now we have to prove that CQ is a bisector of∠BCD. For which we will prove that Q divides BD in the ratio BC: DC.

In ∆ABC , BP the bisector of∠ABC

Therefore ,
AB/BC = AP/PC...................(1)

In ∆ ACD, DP bisector of ∠ADC

AD/DC= AP/PC.....................(2)

From eq. 1 & 2

AB/BC = AD/DC

AB/AD = BC/DC...............(3)

In ∆ABD , AQ is the bisector of∠BAD. (By construction)

AB/AD= BQ/DQ.................(4)

From eq. 3 & 4

BC/ DC=BQ/DQ

Thus , in ∆CBD, Q divides BD in the ratio CB: CD.
Therefore CQ is the bisector of∠BCD.

Hence, bisectors of ∠BAD & ∠BCD meet on the diagonal BD..

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