Prove that if the altitude and median of a triangle form equal angles with sides then the triangle is right.

Question

Problem statement:

Prove that if the altitude and median drawn from the same vertex of a nonisosceles triangle lie inside the triangle and form equal angles with its sides, then this is a right triangle.

After many attempts, I came up with this: let $ABC$ be the triangle where $CH$ is the altitude, $CM$ is the median, and name the angles $ACH = MCB = \theta$, $CAH = \alpha$, $CBM = \beta$ and $HCM = \gamma$. Using the sine theorem in the triangle $CMB$ we get $$\frac{MB}{\sin(\theta)} = \frac{MC}{\sin(\beta)},$$ while using the sine theorem in the triangle $ACM$ we get $$\frac{MC}{\sin(\alpha)} = \frac{MA}{\sin(\theta + \gamma)}.$$ Combining these equations I found $$\sin(\alpha) \sin(\theta) = \sin(\beta) \sin(\theta + \gamma).$$ Applying the identity $$\sin(a)\sin(b) = \frac{\cos(a-b) - \cos(a+b)}{2}$$ I concluded $$\cos(\alpha - \theta) = \cos(\theta + \gamma - \beta).$$ Since all angles are within $[0, \pi]$ range I concluded $$\alpha - \theta = \theta + \gamma - \beta,$$ thus $2 \theta + \gamma = \alpha + \beta$. From the original triangle we know $$2 \theta + \gamma + \alpha + \beta = \pi,$$ and combining the equations proves the triangle is right.

Is this correct? Is there a synthetic way to do it?

Answer 1

Extend median $\overline{AM}$ of $\triangle ABC$ to meet the circumcircle at $M'$, and let $D$ be the foot of the altitude from $A$. (Note that $D$ and $M$ are distinct —and, thus, $\overline{AM'}\not\perp\overline{BC}$— because we assume $\triangle ABC$ is non-isosceles.)

Since $\angle M' \cong\angle B$ (by the Inscribed Angle Theorem), we conclude $\angle ACM'\cong\angle ADB=90^\circ$, so that $\overline{AM'}$ is a diameter. Now, a diameter can only bisect a non-perpendicular chord at its own midpoint (aka, the circumcenter); consequently, $M$ is that circumcenter, $\overline{BC}$ is a diameter, and $\angle BAC$ is a right angle. $\square$

Answer 2

It can also be proved using geometry only.

1] Through $M$, draw $MX \parallel BC$ cutting $AC$ at $X$. From that, we have

• (1.1) $\beta = \beta’$; and
• (1.2) $AX = XC$.

2] Through $X$, draw $XYZ \parallel AB$ cutting $CH$ at $Y$ and $CM$ at $Z$. From that, we have

• (2.1) $XY$ is the perpendicular bisector of $CH$; and
• (2.2) $Z$ is the midpoint of $CM$.

(2.1) + (1.1) implies $\alpha’ = \alpha = \beta = \beta’$. Since $H$ and $M$ are distinct ($\triangle ABC$ is not isosceles), the inscribed angle theorem means $XHMC$ is a cyclic quadrilateral. This also gives us that $\angle MXC = \angle MHC = 90^\circ$.

Then since $MX \parallel BC$, also $\angle ACB = 90^\circ$ as required.

Answer 3

Trigonometric Approach

We are given that $\angle MAC=\angle BAP$ and since $\triangle BPA$ is a right triangle, $\angle BAP=\frac\pi2-B$. Therefore, $$ \angle MAC=\frac\pi2-B\tag1 $$ and, since $\angle MAC+\angle BAM=A$, we have $$ \angle BAM=A+B-\frac\pi2\tag2 $$ Since $M$ is the bisector of $\overline{BC}$, the areas of $\triangle BAM$ and $\triangle MAC$ are equal. Thus, $$ \begin{align} \overbrace{\frac12cm\sin\left(A+B-\frac\pi2\right)}^{\left|\triangle BAM\right|} &=\overbrace{\frac12bm\sin\left(\frac\pi2-B\right)}^{\left|\triangle MAC\right|}\tag{3a}\\ c\cos(C)&=b\cos(B)\tag{3b}\\[3pt] c\sin(B)&=b\sin(C)\tag{3c}\\[3pt] \sin(2B)&=\sin(2C)\tag{3d} \end{align} $$ Explanation:
$\text{(3b)}$: apply $A+B-\frac\pi2=\frac\pi2-C$ and multiply by $\frac2m$
$\text{(3c)}$: Law of Sines
$\text{(3d)}$: cross multiply $\text{(3b)}$ and$\text{(3c)}$ and multiply by $\frac2{bc}$

Since $\triangle ABC$ is not isosceles, $\text{(3d)}$ implies that $2B=\pi-2C$; which, since $A=\pi-B-C$, means $$ \bbox[5px,border:2px solid #C0A000]{A=\frac\pi2}\tag4 $$

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Geometric Approach

This is an approach similar, but slightly different, to that in Mick's answer.

Start with the diagram above. We are given that $\angle MAC$ is equal to $\angle BAP$, which is $\frac\pi2-B$ since $\angle APB$ is a right angle.

Add $N$ and $R$, the midpoints of $\overline{AC}$ and $\overline{AB}$ respectively. Then add the segments from $R$ to $N$, $M$, and $P$.

Add the circle with $\overline{AM}$ as its diameter. Since $\angle APM$ is a right angle, $P$ sits on this circle. Since $\overline{RN}$ is parallel to $\overline{BC}$, $\overline{RN}$ is also perpendicular to $\overline{AP}$.

Since $\triangle ARN$ is similar to $\triangle ABC$ with a linear ratio of $\frac12$, $Q$, the intersection of $\overline{AP}$ and $\overline{RN}$, bisects $\overline{AP}$. Thus, $\angle RPQ$ is also $\frac\pi2-B$.

Since $\overline{RM}$ is parallel to $\overline{AC}$, $\angle RMA$ is equal to $\angle MAC$, which equals $\frac\pi2-B$.

The locus of points at which $\overline{AR}$ subtends an angle of $\frac\pi2-B$ is the circle which contains $A$, $R$, $P$, and $M$. This is the same circle that was added above since it contains $A$, $P$, and $M$. Thus, $\angle ARM$ is a right angle, and since $\overline{RM}$ is parallel to $\overline{AC}$, so is $A$.