Angle bisectors of triangle are concurrent

Angle bisectors of internal angle of a triangle are concurrent

1. Three Angle Bisector of A Triangle are Concurrent Created by Laurado Rindira Sabatini

2. Introduction Prove that three angle bisectors of the internal angles of a triangle are concurrent The three angle bisectors; AD, BE, and FG are concurrent because the point C is on all of the angle bisectors.

3. Given ∆ ABC Construct the angle bisectors of ⦟ABC and ⦟ ACB which intersect at the point O Proof

4. Construct perpendicular line from O to AB, from O to AC, and from O to BC No Proof Reason 1 ∆ DOB ≈ ∆ FOB Angle-Side-Angle 2 OD = OF Result of step 1 3 ∆ EOC ≈ ∆ FOC Angle-Side-Angle 4 OE = OF Result of step 3 5 OD = OE = OF Steps 2 and 4

5. Construct segment AO

6. No Proof Reason 6 ∆ ADO is right triangle AB ⊥ OD 7 ∆ AEO is right triangle AC ⊥ OE 8 OD = OE Step 5 9 AO = AO (The hypotenuse of ∆ ADO and ∆ AEO respectively ) Reflective property 10 ∆ ADO ≈ ∆ AEO Step 8 and 9 11 m ∠ DAO = m ∠ EAO Step 10 12 AO is the angle bisector Step 11 13 O lies on the angle bisector of AO, BO, and CO Constructed and step 12 14 The three angle bisectors of a triangle are concurrent Step 13 Proved

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