## pratu043 3 years ago ABC and DBC are two triangles on the same base BC such that A and D lie on the opposite sides of BC, AB = AC and DB = DC. Show that AD is the perpendicular bisector of BC.

1. pratu043

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2. mukushla

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3. pratu043

ok

4. mukushla

1...show that triangle ABD=triangle ACD......it gives BE=EC 1...show that triangle ABE=triangle AEC......it gives <AEB=<AEC=90

5. pratu043

I have shown that triangle ABD = ACD, but I don't know how to do the other one.

6. mukushla

triangle ABE=triangle AEC because 1...AC=AB 2...BE=EC 3...AE=AE

7. pratu043

But we have to prove that BE = EC right?

8. pratu043

Perpendicular bisector.

9. mukushla

u've shown that triangle ABD=triangle ACD am i right?

10. pratu043

yes.

11. pratu043

sorry if I take a little long to reply.

12. mukushla

np....:)

13. mukushla

well it gives BE = EC

14. pratu043

But how?

15. pratu043

I mean, its not a side of ABD or ACD.

16. mukushla

u r right ... thats not a complete proof... lets think again...

17. mukushla

we need to show that triangle ABE=triangle AEC...........

18. mukushla

try to find 2 angles equal...and one side equal for that 2 triangle

19. pratu043

AB = AC

20. pratu043

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21. pratu043

so that you don't need to scroll up too much.

22. mukushla

lol.........:)

23. mukushla

we proved that triangle ABD=triangle ACD so that... <BAE=<CAE and what is the third condition?

24. pratu043

AE is common so triangle AEB = AEC.

25. mukushla

we are done

26. pratu043

But how does that prove that AD is the perpendicular bisector?

27. pratu043

We should prove that AD bisects BC and angle AEB = 90 degrees.

28. mukushla

it provides BE=EC and <AEB=<AEC but we know that <AEB+<AEC=180

29. pratu043

Thanks a lot mukushla!!!

30. mukushla

yw......:)