Definition 3.3.1. Median.
A line is a median if and only if it connects a vertex of a triangle to the midpoint of the opposing side.
Section 3.3 Concurrent
Subsection 3.3.1 Explore
Geogebra will be helpful for performing these experiments. Be as detailed as you can with your conjectures.
Checkpoint 3.3.2.
Use the Geogebra example in Figure 3.3.3 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?
Instructions.
Drag the vertices of the triangles around and see what is always true about the perpendicular bisectors.
Checkpoint 3.3.4.
Use the Geogebra example in Figure 3.3.5 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?
Instructions.
Drag the vertices of the triangles around and see what is always true about the medians.
Checkpoint 3.3.6.
Use the Geogebra example in Figure 3.3.7 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?
Instructions.
Drag the vertices of the triangles around and see what is always true about the angle bisectors.
Checkpoint 3.3.8.
Use the Geogebra example in Figure 3.3.9 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?
Instructions.
Drag the vertices of the triangles around and see what is always true about the altitudes.
Checkpoint 3.3.10.
Construct \(\triangle ABC.\) Construct \(\triangle XYZ\) such that \(X-B-Y,\) \(Y-C-Z,\) \(Z-A-X,\) and \(\overline{XY} \parallel \overline{AC},\) \(\overline{YZ} \parallel \overline{AB},\) \(\overline{ZX} \parallel \overline{BC}.\) Construct the perpendicular bisectors of \(\triangle XYZ.\) What appears to be true of these with respect to \(\triangle ABC.\)
Subsection 3.3.2 Prove
Lemma 3.3.11.
Consider three points \(A,B,C\) with \(\ell_1\) and \(\ell_2\) the perpendicular bisectors of \(\overline{AB}\) and \(\overline{BC}\) respectively. Let \(M_2=\ell_2 \cap \overline{BC}.\) Show \(\ell_1 \parallel \ell_2\) implies the existence of \(D=\ell_2 \cap \stackrel{\longleftrightarrow}{AB}\) such that \(A,\) \(B,\) and \(D\) are collinear and \(\angle BDM_2\) is a right angle.
Theorem 3.3.12. Perpendicular bisectors.
Prove the conjecture about the perpendicular bisectors.
Theorem 3.3.13. Circumcenter.
Three, non-colinear points uniquely determine a circle.
Lemma 3.3.14.
Two medians intersect at a point \(2/3\) of the way down both medians.
Theorem 3.3.15. Medians.
Prove the conjecture about the medians.
Theorem 3.3.16.
A point is on the angle bisector of an angle if and only if it is equidistant from both sides of the angle.
Theorem 3.3.17. Angle bisectors.
Prove the conjecture about the angle bisectors.
Theorem 3.3.18. Incenter.
For each triangle there exists a circle inside and tangent to all three sides.
Theorem 3.3.19. Altitudes.
Prove the conjecture about the altitudes.
