Say the 4 smaller circles are Q,R,S,T.

R is tangent to Q and S

The quadrilateral is ABCD.

The incircle is P.

All the 5 circles are named after their respective centerpoints.

Incircle P is tangent to AB at point E. So, PE is perpendicular to AB.

Incircle P is tangent to BC at point F. So, PF is perpendicular to BC.

Circle R is tangent to side AB at point G, and tangent to BC at point H.

Circle Q is tangent to side AB at point J....

Circle S is tangent to side BC at point K....

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Consider circles R and Q.

If R and Q are tangent to each other, then their point of tangency must be along the apothem (inradius) PE. Call that point, X.

Then line RXQ is a straight line segment.

And RQ is perpendicular to PE.

If circlpe Q is tangent to AB, then radius QJ is perpendicular to AB.

Likewise, if circle R is tangent to AB, then radius RG is perpendicular to AB also.

Hence, since RG, PE, QJ are all perpendicular to AB, then RG, PE, QJ are parallel.

Line GEJ is a straight line segment, because points G,E,J are on AB.

So, GJ is perpendicular to RG, PE, QJ.

Hence, RQJG is a rectangle.

In circle R, RG = RX. Both are radii of R.

In circle Q, QJ = QX. Both are radii of Q.

In rectangle RQJG, RG = XE = QJ.

So, RG = QJ

Therefore, circle R is congruent to circle Q, because they have equal radii.

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Repeat that for circles R and S.

It will be the same. Circles R and S are congruent too.

Blah, blah, blah, all four smaller circles are congruent.

And ABCD is a square---cannot be otherwise.