GIVEN: TRIANGLE ABC,AD bisects <BAC,and AE=ED, PROVE:AE/AC=BD/BC
To prove that AE/AC = BD/BC given triangle ABC, where AD bisects angle BAC and AE = ED, we can use the Angle Bisector Theorem and the Segment Addition Postulate.
Angle Bisector Theorem: In a triangle, if a line segment bisects an angle, it divides the opposite side into two segments that are proportional to the adjacent sides of the angle.
Segment Addition Postulate: If three points A, B, and C are collinear and B is between A and C, then AB + BC = AC.
Here's a step-by-step proof:
Step 1: Draw the triangle ABC and the angle bisector AD.
Step 2: By the Angle Bisector Theorem, we know that AE/ED = AC/CD.
Step 3: Since AE = ED, we can substitute AE for ED in the equation from step 2: AE/AE = AC/CD.
Step 4: Simplify the equation from step 3: 1 = AC/CD.
Step 5: Multiply both sides of the equation from step 4 by CD: CD = AC.
Step 6: Apply the Segment Addition Postulate to the segment AC, so AC = AD + DC.
Step 7: Substitute the value from step 6 into the equation from step 5: CD = AD + DC.
Step 8: Rearrange the equation from step 7 to isolate AC: AC = CD - AD.
Step 9: Now, let's look at the triangle ABD. Using the Angle Bisector Theorem, we know that AE/EB = AD/DB.
Step 10: Substitute the value AE from the equation in step 3: AE/EB = AD/DB.
Step 11: Since AE = ED, we can substitute AE for ED in the equation from step 10: AE/AE = AD/DB.
Step 12: Simplify the equation from step 11: 1 = AD/DB.
Step 13: Multiply both sides of the equation from step 12 by DB: DB = AD.
Step 14: Apply the Segment Addition Postulate to the segment BD, so BD = AB + AD.
Step 15: Substitute the value from step 14 into the equation from step 13: DB = AB + AD.
Step 16: Rearrange the equation from step 15 to isolate BD: BD = DB - AB.
Step 17: Now, let's go back to the original equation to be proven, AE/AC = BD/BC.
Step 18: Substitute the values from steps 8 and 16 into the equation from step 17: AE/(CD - AD) = (DB - AB)/BC.
Step 19: Cross-multiply the equation from step 18 to eliminate the denominators: AE(BC) = (DB - AB)(CD - AD).
Step 20: Expand both sides of the equation in step 19: AE * BC = DB * CD - DB * AD - AB * CD + AB * AD.
Step 21: Rearrange the terms in the equation from step 20: AE * BC = DB * CD + AB * AD - AB * CD - DB * AD.
Step 22: Rearrange and group the terms on the right-hand side of the equation from step 21: AE * BC = (DB * CD - DB * AD) + (AB * AD - AB * CD).
Step 23: Factor out the common terms in each group on the right-hand side of the equation from step 22: AE * BC = DB(CD - AD) + AB(AD - CD).
Step 24: Apply the commutative property of addition to rearrange the terms in the equation from step 23: AE * BC = DB(AD - CD) + AB(AD - CD).
Step 25: Factor out (AD - CD) from each term on the right-hand side of the equation from step 24: AE * BC = (DB + AB)(AD - CD).
Step 26: Since (AD - CD) = -1 * (CD - AD), we can rewrite the equation from step 25: AE * BC = (DB + AB)(CD - AD).
Step 27: Since AE = ED, substitute ED for AE in the equation from step 26: ED * BC = (DB + AB)(CD - AD).
Step 28: Apply the distributive property to the right-hand side of the equation from step 27: ED * BC = DB * (CD - AD) + AB * (CD - AD).
Step 29: Rearrange the terms on the right-hand side of the equation from step 28: ED * BC = DB * CD - DB * AD + AB * CD - AB * AD.
Step 30: Rearrange and group the terms on the right-hand side of the equation from step 29: ED * BC = (DB * CD + AB * CD) - (DB * AD + AB * AD).
Step 31: Factor out CD from the terms grouped together on the right-hand side in step 30: ED * BC = (DB + AB) * CD - (DB + AB) * AD.
Step 32: Factor out (DB + AB) from both terms on the right-hand side of the equation from step 31: ED * BC = (DB + AB)(CD - AD).
Step 33: Since (CD - AD) = -(AD - CD), we can rewrite the equation from step 32: ED * BC = (DB + AB)(AD - CD).
Step 34: Since DB + AB = BA + AD, substitute BA + AD for DB + AB in the equation from step 33: ED * BC = (BA + AD)(AD - CD).
Step 35: Apply the commutative property of multiplication to rearrange the terms in the equation from step 34: ED * BC = (AD + BA)(AD - CD).
Step 36: Since AD + BA = AB + AD, we can rewrite the equation from step 35: ED * BC = (AB + AD)(AD - CD).
Step 37: Since AB + AD = BD, substitute BD for AB + AD in the equation from step 36: ED * BC = BD(AD - CD).
Step 38: Divide both sides of the equation from step 37 by BC to isolate the term BD: ED = (BD * (AD - CD))/BC.
Step 39: Since AE = ED, we can substitute ED for AE in the equation from step 38: AE = (BD * (AD - CD))/BC.
Step 40: Multiply both sides of the equation from step 39 by BC to isolate the term AE: AE * BC = BD * (AD - CD).
Step 41: Divide both sides of the equation from step 40 by (AD - CD) to isolate the term BD: (AE * BC)/(AD - CD) = BD.
Step 42: Divide both sides of the equation from step 41 by BC to isolate the term AE: AE/(AD - CD) = BD/BC.
Step 43: Recognize that (AD - CD) = AC, so we can replace (AD - CD) with AC in the equation from step 42: AE/AC = BD/BC.
Therefore, AE/AC = BD/BC is proven using the Angle Bisector Theorem and the Segment Addition Postulate.