Proof that there is a point inside an acute-angled triangle so that perpendiculars dropped to the sides will form vertices of equilateral triangle

Question

There is a character limit to the title length and the task may not be clear. So here is a little bit longer task description.

Task description:

You are given an acute-angled triangle $ABC$. Proof that there is a point $O$ inside $ABC$ so if you drop perpendiculars $OD, OE, OF$ to the sides: $$D \in AB,\; E \in BC,\; F \in AC,\; OD \bot AB,\; OE \bot BC,\; OF \bot AC$$ then $DEF$ is an equilateral triangle.

Questions I found useful:

Inscribe an equilateral triangle inside a triangle
Three lines are given. Find three points on these lines, one point on each line, that are vertices of an equilateral triangle

Current idea:

1. Choose $D' \in AB$
2. Inscribe an equilateral triangle $D'E'F'$ inside $ABC$ (Problem here is that sometimes $E'$ and $F'$ are outside of $BC$ and $AC$ but let's assume they are inside)
3. Draw perpendiculars to $AB$ through $D'$, to $BC$ through $E'$, to $AC$ through $F'$
4. Let's say perpendiculars through $E'$ and $F'$ intersect at $O'$
5. It seems that if we move $D'$ from $A$ to $B$ then the $O'$ will be to the $B$ side from perpenducilar through $D'$ at the beginning and to the $A$ side at the end (I can't proof this). So there will be the moment when perpendicular through $D'$ will go through $O'$.
6. At this point $O'$ seems to be $O$ we are searching for but I can't proof that $O'$ will be inside a triangle $ABC$.

Will be pleased for any help (ideas/links to similar questions)

UPDATE: Found point $O$ in Encyclopedia of Triangle Centers (https://faculty.evansville.edu/ck6/encyclopedia/ETC.html):
X(15) = 1st ISODYNAMIC POINT
The pedal triangle of X(15) is equilateral.

UPDATE 2: Here is how to find $O$ with straightedge and compass: https://mathworld.wolfram.com/IsodynamicPoints.html