Question

If mABD = mACD = 90° then show That point A, B, C and D are concyclic​

Answer 1

$\huge \mathcal{ \fcolorbox{cyan}{black}{ \pink{Answer : - }}}$

✪ In order to prove that the points B,C,E and D are concyclic, it is sufficient to show that ∠ABC+∠CED=180$0 and ∠ACB+∠BDE=1800.

In △ABC, we have

 AB=AC and AD=AE

⇒ AB−AD=AC−AE

⇒ DB=EC

Thus, we have

  AD=AE and DB=EC 

⇒   DBAD=ECAE

⇒   DE∣∣BC          [By the converse of Thale's Theorem]

⇒   ∠ABC=∠ADE          [Corresponding angles]

⇒   ∠ABC+∠BED=∠ADE+∠BDE           [Adding ∠BDE both sides] 

⇒  ∠ABC+∠BDE=1800

⇒   ∠ACB+∠BDE=1800           [∵AB=AC∴∠ABC=∠ACB]

Again,              DE∣∣BC

⇒    ∠ACB=∠AED

⇒   ∠ACB+∠CED=∠AED+∠CED            [Adding ∠CE on both sides]

⇒  ∠ACB+∠CED=1800

⇒  ∠ABC