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prove that the sum of the perpendiculars from a point on the base of an isosceles triangle is equal to the altitude of the vertex of a base angle.

Please see picture for clear understanding of the question

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    Why would you do Euclidean geometry without similarity? Are you saying the problem requires this or you are just uncomfortable using similarity? – 2017-02-20
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    @rschwieb Because you do not need it for this problem. Such problems are given to students while they learn geometry with proofs. They start with concepts such as main axioms and basic theorems for angles, perpendiculars, then congruent triangles and properties,isosceles, equilateral and some right triangles. This is when this problem appears. Later, students learn about midpoints, midsegments, intercept theorems, circle properties, then they arrive at similar triangles, and after that Pythagoras' theorem, right triangles and trigonometry... you get the idea. – 2017-02-20
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    @Futurologist Well, it is obvious to speculate that "maybe it is not needed" in any situation like this. Unfortunately, if you are wrong and the poster's attitude is just "fear of similarity," they are missing out on the simple solution via similarity. Perhaps now we should wait for the person whom the question was directed at to answer it... – 2017-02-20

3 Answers 3

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Draw a line $l$ through $P$ parallel to $AB$ .Let $E$ be the point of intersection of $l$ and $FC$. Now $FSPE$ is a parallelogram thus $SP = FE$ (1).Now let $G$ be the point of intersection of $l$ and $AC$,then $\angle GPC=\angle ABC=\angle ACB$ and thus $PGC$ is isosceles and thus $CE=PQ$ (2),as they are both heights of the isosceles triangle to the equal sides.

From (1) and (2) the original statement is derived.

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Hint: Draw a line parallel to $AB$ passing through point $P$ and let it intersect $CF$ at point $E$ and $CA$ at point $R$. What triangle is $CPR$ and what kind of segments are $PQ$ and $CE$ in this triangle? Do you see it?

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Let $\ell=AB=AC$. The area of $ABC$ is given by $$\ell\cdot CF = PS\cdot\ell+PQ\cdot \ell $$ hence it follows that $CF=PS+PQ$. Dissection works just fine.
As an alternative, you may invoke Viviani's theorem.