Easy Compass Construction Problem

Question

Here is a tricky compass and straightedge construction problem.

Given triangle $\triangle ABC$ and point $D$ on segment $\overline{AB}$, construct point $P$ on line $\overleftrightarrow{CD}$ such that $\angle APB = \angle BAC$.

This configuration appears frequently in Olympiad geometry problems and the diagram is impossible to draw precisely unless you know how to do the construction.

Answer 1

I misread the question the first time I answered this, so let me try again.

The condition that you gave $\angle APB = \angle BAC$ is the degenerate case of the "two angles that inscribe the same arc are equal" theorem. More specifically, if you were to draw the circumcircle of $\triangle APB$, you would see that the tangent to $A$ is $AC$. Also, we know that $$\angle APB = 180^\circ - \angle AID = \angle AIC$$ where $I$ is the intersection of the circumcircle of $APB$ with $BC$. This implies that $$\triangle ABC \sim \triangle IAC.$$ The construction of $I$ isn't that difficult (just copy $\angle B$ onto side $AC$), and everything can be reversed to find $P$.

Answer 2

Construct isosceles triangle QAB with QA equals QB and angle QAB equals 90- BAC/2. The circumcircle of QAB is the locus of points X ( above the line AB) such that angle AXB equals BAC.. The desired point P is the intersection of this circle with CD.

Answer 3

There are 2 possible cases – $P$ can be outside or inside of $\triangle ABC$. Maybe that is why the OP claims that this is trick problem.

Case-1 ($P$ is outside, easier to start with for my $ABC$)

1) Draw line $g$, the perpendicular bisector of $AB$.

2) Draw line $h$, through $A$ and perpendicular to $AC$, cutting $g$ at $O$.

3) Draw circle $k$ using $O$ as center and $OA$ as radius.

4) Let circle $k$ cuts $CD$ (extended) at $P$.

Proof: By angles in alternate segment, $\alpha = \beta$.

Case-2 ($P$ is inside triangle ABC.)

[Continuing from the above]

1) Locate $P’$, the mirror refection of $P$ about $AB$. It should be clear that $\theta = \alpha$.

2) Draw circle $m$, passing through $A, B, P’$, cutting $CD$ at $P”$.

Proof: By angles in the same segment, $\phi = \theta$.

Result follows.