ABC is an inscribed triangle and chord AD bisects angle A and cuts BC at P. Prove: (AD)/(CD)=(AC)/(PC). GIven: Inscribed △ ABC, chord AD bisects PSK∠ A and cuts BC at P; chord CD, Prove: (AD)/(CD)=(AC)/(PC).

Braeden Valenzuela

Braeden Valenzuela

Open question

2022-08-20

ABC is an inscribed triangle and chord AD bisects angle A and cuts BC at P. Prove: ADCD=ACPC.
GIven: Inscribed ABC, chord AD bisects A and cuts BC at P; chord CD
Prove: ADCD=ACPC.
Statements
1.(see above)
2. D=D
3.
4. BCD=DAB
5.
6. ACDCPD
7. ADCD=ACPC
Reasons
1.Given
2.
3.Defintion of angle bisectors
4.
5.Substitution
6.
7.

Answer & Explanation

Nathalie Foster

Nathalie Foster

Beginner2022-08-21Added 7 answers

Step1
D=D
Here angle D is equal to itself, that is reflexive property.
Given, AD bisects angle A, so by definition of angle bisector
CAD=DAB
Step2
From the picture, we see that angle BCD and angle DAB are two inscribed angles with the same base points on the circle.
So they are equal.
BCD=DAB
If we substitute angle DAB by angle BCD in
CAD=DAB
We get:
BCD=CAD
So triangle ACD and triangle CPD has two pairs of angles equal. So they are similar by AA property.

Corollary 57-1 If two angles of one triangle are equal respectively to two angles of another, then the trianglesare similar. (a.a.)
So,
ADCD=ACPC
by C.S.S.T.P, - corresponding sides of similar triangles are proportional.
Answer:
2) Reflexive property
3) CAD=DAB
4)Inscribed angles with same base points
5) BCD=CAD
6) a.a
7) C.S.S.T.P.

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