Euler Line in a Triangle with a 60 Degrees Angle

WLOG, choose $B=(0,\sqrt{3}),\;$ $P=(-1,0),\;$ $Q=(1,0).\;$ Let $A=(x,y).\;$ We have: $\overrightarrow{PA}=k\overrightarrow{BP},\;$ $k\gt 0,\;$ implying

$(x+1)\overrightarrow{i}+y\overrightarrow{j}=-k\overrightarrow{i}-k\sqrt{3}\overrightarrow{j}.$

From here, $A=(-(1+k),-k\sqrt{3}).\;$ Similarly, $C=(1+m,-m\sqrt{3},\;$ with $m\gt 0.\;$ From $AP+CQ=PQ\;$ we obtain $k+m=1,\;$ and from here, $-k\sqrt{3}-m\sqrt{3}+\sqrt{3}=0.\;$ This means that centroid $G\;$ of $\Delta ABC\;$ is $G=\displaystyle\left(\frac{m-k}{3},0\right),\;$ and, thus $G\in PQ.$

Let $H=(\alpha,0)\in PQ\;$ such that $BH\perp AC.\;$ This gives $\overrightarrow{BH}=\alpha\overrightarrow{i}-\sqrt{3}\overrightarrow{j}\;$ and $\overrightarrow{AC}=3\overrightarrow{i}-(k-m)\sqrt{3}\overrightarrow{j}.\;$ But $\overrightarrow{BH}\cdot\overrightarrow{AC}=0,\;$ so that $3\alpha-3(k-m)=0.\;$ In other words, $H=(k-m,0),\;$ implying that the slope of $AH\;$ is

$\displaystyle\frac{k\sqrt{3}}{2k+1-m}=\frac{k\sqrt{3}}{3k}=\frac{1}{\sqrt{1}}.$

But the slope of $BC\;$ is $-\sqrt{3},\;$ hence $AH\perp BC,\;$ making $H\;$ the orthocenter of $\Delta ABC.\;$ Since $H\in PQ,\;$ this completes the proof.

Since $AP+CQ=PQ,\;$ there is point $H\in PQ\;$ such that $AP=HP\;$ and $CQ=HQ.\;$

Triangles $APH\;$ and $CQH\;$ are isosceles with the apex angle $120^{\circ}\;$ and base angles of $30^{\circ}.\;$ Since $\angle HAP+\angle ABC=30^{\circ}+60^{\circ}=90^{\circ},\;$ $AH\perp BC.\;$ Similarly, $CH\perp AB,\;$ making $H\;$ the orthocenter of $\Delta ABC.\;$ Note that $H\in PQ.$

Since $AP+CQ=PQ,\;$ there is point $O\in PQ\;$ such that $AP=OQ\;$ and $CQ=OP.\;$ Triangles $APO\;$ and $CQO\;$ are equal, making $\Delta AOC\;$ isosceles. Thus, $O\;$ lies on the perpendicular bisector of $AC.\;$ We'll show that it also lies on the perpendicular bisectors of $AB\;$ and $BC.$

For example, let $P'\in AB\;$ satisfy $BP'=AP.\;$ Then, since $BP=PQ,\;$ $PP'=PO,\;$ so that $\Delta POP'\;$ is equilateral, and $O\;$ lies on the perpendicular bisector of $PP'.\;$ By the construction, this is also the perpendicular bisector of $AB.\;$ Side $BC\;$ is treated similarly. It follows that $O\;$ is the circumcenter of $\Delta ABC,\;$ implying that $PQ\;$ is indeed its Euler's line.

The above confirms an old result that the Euler line in a triangle with an angle of $60^{\circ}\;$ cuts off an equilateral triangle. In such a triangle, the condition of the problem is both necessary and sufficient. This is because, by continuity, among all lines $PQ\;$ that cut off an equilateral $\Delta PBQ,\;$ only one may satisfy the condition $AP+CQ=PQ.$

The problem has been kindly posted at the CutTheKnotMath facebook page, along with his solution (Solution 1), by Leo Giugiuc. Leo credits the problem to Kadir Altintas (Turkey) and Ruben Dario (Peru).