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For a given triangle $ABC$, how to construct a point $P$ such that $PA \colon PB \colon PC = 1 \colon 2 \colon4$?

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Given points A and B, the locus of points P such that PA:PB = 1:2 (P is twice as far from B as it is from A) is a circle, and similarly for locus PA:PC = 1:4. So draw the two circles, and P is at an intersection of the circles. Note that you may get 0, 1, or 2 possible points for P. In the diagram below I show a situation giving 2 points $P_1$ and $P_2$.

Construct Given Ratios from Triangle Vertices

Here is an overview of the construction:

1) Construct point $H$ on $\overline {AB}$ such that $AH = \frac{1}{3}AB$.

2) Construct point $J$ on ray $\overrightarrow {BA} $ (not on $\overline {AB}$) such that $AJ = AB$.

3) Construct the circle that has $\overline {HJ}$ as a diameter.

4) Construct point $U$ on $\overline {AC}$ such that $AU = \frac{1}{5}AC$.

5) Construct point $W$ on ray $\overrightarrow {CA} $ (not on $\overline {AC}$) such that $AW = \frac{1}{3}AC$.

6) Construct the circle that has $\overline {UW}$ as a diameter.

7) Mark point $P$ at an intersection of the circles.

ADDED:

I just realized that it would be easier to construct the 1:2 circle between A and B and the 1:2 circle between B and C. Oh, well, what I showed above also works.

ADDED LATER:

I just had to add a diagram of the better construction using two 1:2 circles. In the diagram below, $AH = \frac13AB$, $AI=AB$, $BQ=\frac13BC$, $BM=BC$, and $\overline {HI}$ and $\overline {MQ}$ are diameters of circles $p$ and $u$ respectively.

Better Construction of Given Ratios from Triangle Vertices

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Here is a solution:

Orient the triangle such that AC lies on the x-axis and B lies on the y-axis. Then, we have that $A = (-a,0), B = (0,b), C = (0,c)$. Draw circles around each of these points with a radius $r$, $2r$ and $4r$.

If these circles are to meet at a point $(x, y)$, then we have the following simultaneous equations:

$$(x+a)^2 + y^2 = r^2$$ $$x^2 + (y-b)^2 = (2r)^2$$ $$ (x-c)^2 + y^2 = (4r)^2$$

There are three equations and three variables. You can now solve it to obtain, $x$ and $y$ (and $r$ too which will be helpful if you want to physically construct it with a compass).

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