3.3: Concurrent
- Page ID
- 89849
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3.3.1 Explore
Geogebra will be helpful for performing these experiments. Be as detailed as you can with your conjectures.
Use the Geogebra example in \(\PageIndex{1}\) to experiment with the relationship of the three perpendicular bisectors of a triangle. Move the vertices of the triangle around. What remains true about the perpendicular bisectors?
Use the Geogebra example in \(\PageIndex{2}\) to experiment with the relationship of the three medians of a triangle. Move the vertices of the triangle around. What remains true about the medians?
Use the Geogebra example in \(\PageIndex{3}\) to experiment with the relationship of the three angle bisectors of a triangle. Move the vertices of the triangle around. What remains true about the angle bisectors?
Use the Geogebra example in \(\PageIndex{4}\) to experiment with the relationship of the three altitudes of a triangle. Move the vertices of the triangle around. What remains true about the altitudes?
Construct △ABC. Construct △XYZ such that X-B-Y, Y-C-Z, Z-A-X and XY ll AC, YZ ll AB, ZX ll BC. Construct the perpendicular bisectors of △XYZ. What appears to be true of these with respect to △ABC.
3.3.2 Prove
Consider three points A, B, C with ℓ1 and ℓ2 the perpendicular bisectors of AB and BC respectively. Let M2 = ℓ2 ∩ BC. Show ℓ1 ll ℓ2 implies the existence of D = ℓ2 ∩ AB such that A, B and D are collinear and ∠BDM2 is a right angle.
Prove the conjecture about the perpendicular bisectors.
Three points uniquely determine a circle.
Two medians intersect at a point 2/3 of the way down both medians.
Prove the conjecture about the medians.
A point is on the angle bisector of an angle if and only if it is equidistant from both sides of the angle.
Prove the conjecture about the angle bisectors.
For each triangle there exists a circle inside and tangent to all three sides.
Prove the conjecture about the altitudes.