while remaining similar to itself all the time. Then there is a point O, rigidly attached to the triangle,

remaining fixed all the time. Every point rigidly attached to the triangle glides also on a fixed line.

By the expression

the varying ABC. Let {A

ABC

{L

of O. In fact, relative to ABC, O is the point viewing the sides under the respective angles: BC under π-A

π-C

shows that O has fixed position with respect to A

basic for this subject, discussion in Similarly_Rotating.html .

Point O is called a

its vertices glide correspondinly on one of the three lines L

restricted to move so that it always circumscribes a fixed triangle while remaining all the time similar to itself and each one of its vertices

is viewing a certain side of the fixed triangle under a constant angle. In this case it was shown that every point of the moving triangle,

which is rigidly attached to it, moves along a circle which passes through a fixed point O. This point was also called

moving outer triangle) because it coincides with the equal named point considered here. Simply instead to turn the outer triangle around, as

was the case there, here we turn the inscribed triangle around O, fixing the circumscribing triangle defined by the lines L

The difference of the two procedures is reflected in the way they vary all rigidly attached points to the moving triangle. In that case they

move along circles passing through O, in the present case they move along lines.

There are twelve ways to let a similar to ABC triangle glide with its vertices lying all the time on these three lines. Isosceli though

reduce to

intersections of the three lines. The figure below illustrates such a case (see Isodynamic.html and ApolloniusCircles.html ).

The above figure shows the two pivots creating all the various possible positions of an equilateral triangle whose vertices glide on the

three side-lines of triangle ABC.

Next figure illustrates another case, concerning the isosceles right-angled triangle. The six (red) points are all the pivots creating all possible

positions for the isosceles right-angled triangle to be inscribed in ABC. The pivots lie by three on two circles orthogonal to the bundle of

Apollonian circles of ABC. Each Apollonian circle contains two of them which are interchanged by the inversion with respect to the

circumcircle of ABC.

inscribed in an arbitrary quadrilateral (i.e. a square having each vertex on a different side-line of a given quadrilateral).

For this it suffices to do the following:

[1] Single out three sides of the quadrilateral, {AB, BC, DA} say, making the triangle ABJ.

[2] Construct a pivot H of rotation of a right-angled isosceles EFG in ABJ.

[3] Complete EFG to a square EFGI and consider triangle HEI, having H fixed E on line AD and remaining

similar to itself. The locus of I, as HEI turns around H is a line (e).

[4] The intersection point K of (e) with the fourth side, CD, of the qudrilateral is a vertex of the inscribed square.

[5] To complete the square construction build on HK triangle HKL similar to HEI and construct on side KL the

square KLMN.

In general each of the six pivots of rotation of EFG in ABJ, playing the role of H, will deliver a different square and

we will have six solutions. Such a configuration of the six squares inscribed in a quadrilateral can be seen in

Inscribed_Squares_6.html .

Similarly_Rotating.html

Similarity.html

Isodynamic.html

ApolloniusCircles.html

Inscribed_Squares_6.html

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