Given: Triangle ABC with AD bisecting angle BAC, and AE is congruent to ED (both line segments).
Prove: AE/AC = BD/BC.
Statements Reasons
To prove that AE/AC = BD/BC, we can use the Angle Bisector Theorem and the Segment Addition Postulate.
1. Triangle ABC with AD bisecting angle BAC, and AE is congruent to ED (both line segments) - Given.
2. By the Angle Bisector Theorem, we know that AE/EC = AD/DC. This is because AD bisects angle BAC, so according to the Angle Bisector Theorem, the ratio of the segments containing the angle bisector is equal to the ratio of the lengths of the opposite sides. Here, the angle bisector is AD, and the opposite sides are AE and EC.
3. Since AE is congruent to ED, we can substitute ED for AE in the equation from step 2: ED/EC = AD/DC.
4. Using the Segment Addition Postulate, we can express the lengths of the segments in terms of their respective total lengths: AE = AC - EC and ED = DC - EC.
5. Substituting the values from step 4 into the equation from step 3, we get (DC - EC)/EC = AD/DC.
6. Cross multiplying, we have EC*(DC - EC) = AD*EC.
7. Expanding the equation in step 6, we get ECD - EC^2 = ADE.
8. Rearranging the equation in step 7, we have ECD = EC^2 + ADE.
9. Using the Segment Addition Postulate again, we can express the lengths of the segments in terms of their respective total lengths: BD = BC - CD and AC = AE + EC.
10. Substituting the values from step 9 into the original equation of AE/EC = AD/DC, we get (AC - EC)/EC = BD/(BC - CD).
11. Cross multiplying, we have EC*(AC - EC) = BD*EC.
12. Expanding the equation in step 11, we get ACE - EC^2 = BDEC.
13. Rearranging the equation in step 12, we have ACE = EC^2 + BDEC.
14. Comparing the equations in steps 8 and 13, we can see that ECD and BDEC are equal.
15. Therefore, EC^2 + ADE = EC^2 + BDEC.
16. Subtracting EC^2 from both sides, we have ADE = BDEC.
17. Dividing both sides by ACE, we get ADE/ACE = BDEC/ACE.
18. From the Segment Addition Postulate, we know that AE = AC - EC, so ADE/ACE represents AE/AC.
19. Similarly, from the Segment Addition Postulate, we know that BD = BC - CD, so BDEC/ACE represents BD/BC.
20. Therefore, AE/AC = BD/BC, as desired.