Angle bisectors of triangle are concurrent
The three angle bisectors of a triangle are concurrent at a single point. This is proved by first constructing the angle bisectors of any triangle. It is then shown through a series of steps using properties of similar triangles and perpendicular lines that the three angle bisectors intersect at the same point. Specifically, it is shown that two pairs of triangles are similar, meaning the corresponding parts are proportional. This implies that two of the angle bisectors are the same length. It is also shown that the triangles formed at one end of an angle bisector are right triangles. Putting this all together, it is proved that the angle measure of the angles where the bisectors meet the triangle sides are equal, meaning the bisectors intersect at a
Angle bisectors of triangle are concurrent
- 1. Three Angle Bisectors of a Triangle are Concurrent Created by Laurado Rindira Sabatini
- 2. 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
- 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
- 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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