Final exam

Due in class Wednesday, December 21 (hard deadline)

• Answer exactly FIVE (5) of the questions below. If you answer more than five, only the first five will be graded.
• Provide justification to support every claim. You may use (without proving them) theorems/formulas that were stated in class and results that you proved in Homeworks 1 to 4. Reference these results clearly.
• You may use course notes (both your own and uploaded PDFs), the course OpenLab site (including your classmates’ presentation material), and official course textbooks. (If you use textbooks, you may use only results that were stated in class.) You may not use other resources.
• You must work independently; no collaboration allowed. The work you hand in must be your own. You may review the college, department, and class academic integrity policies here.
• Let me know if you have any questions.

1. Let $\triangle ABC$ be a triangle in the plane. Let $D$ be a point anywhere on $\overline{AB}$ and let $E$ be a point anywhere on $\overline{AC}$. Let $F$ be the point where the angle bisectors of $\angle ABE$ and $\angle ACD$ intersect. Prove that $\angle BDC + \angle BEC = 2 \cdot \angle BFC$.
2. Consider a right triangle $\triangle ABC$ with right angle at vertex $C$. Let $D$ be the point on $\overline{AB}$ so that $BC=BD$ and let $E$ be the point on $\overline{AB}$ so that $AC = AE$. Drop the perpendicular from $D$ to $\overline{AC}$ so that it intersects $\overline{AC}$ at the point $F$. Drop the perpendicular from $E$ to $\overline{BC}$ so that it intersects $\overline{BC}$ at the point $G$. Prove that $DF+ EG = DE$.
3. Let $\triangle ABC$ be an isosceles triangle in the plane with $AB = AC$. Let $D$ be any point on $\overline{BC}$. Drop perpendiculars from $D$ to $\overline{AB}$ and $\overline{AC}$. Prove that the sum of the lengths of these perpendiculars does not depend on the position of $D$ on $\overline{BC}$. (Hint: drop the perpendicular from $B$ to $\overline{AC}$.)
4. Consider a rhombus $\square ABCD$ in the plane. Construct a square outside the rhombus with side $\overline{AB}$. Prove that the area of the square is at least the area of the rhombus.
5. Let $\triangle ABC$ be a right triangle with right angle at $C$. Drop the perpendicular from $C$ to $\overline{AB}$ so that it intersects $\overline{AB}$ at $D$. Prove that $(BC)^2 – (AC)^2 = (BD)^2 – (AD)^2$.
6. Consider the circumcircle of $\triangle ABC$ and let $\ell$ be its tangent at $A$. Prove that the angle formed by $\overline{AB}$ and $\ell$ is congruent to the angle of the triangle at $C$. (Hint: there are three cases depending on where the center of the circle lies relative to the triangles’ edges.)
7. Consider a plane $P$ in 3-dimensional space containing points $C$ and $D$. Let $A$ and $B$ be points not on $P$ so that $\overline{AD}$ and $\overline{BC}$ are perpendicular to $P$. Consider a second plane $P’$ through $A$ and perpendicular to $\overline{AB}$ and assume $P$ and $P’$ intersect in the line $\overleftrightarrow{EF}$. Show that $\overline{CD}$ and $\overleftrightarrow{EF}$ are perpendicular in $P$.
8. A cube is constructed inside a right circular cone so that its base lies on the base of the cone and its other four vertices lie on the lateral surface of the cone. Let $h$ be the height of the cone and let $r$ be the radius of its base. Assume that $h = \sqrt{2}$. Prove that the height of the cone is twice the height of the cube.
9. Consider a pyramid with a square base and equilateral triangles for its faces. Prove that the height of the pyramid is equal to half the diagonal of the base.