Question

A. How to determine the side lengths of the enclosed triangle
- Label the three squares S1, S2, S3 and let their side lengths be s1, s2, s3. Also label and fix the orientations/positions of the squares the way they sit on the parcel (for example, which square edge is adjacent to which).
- Put the figure in a coordinate system. A convenient choice is to place one square with one side along the x-axis (for instance S1 from (0,0) to (s1,0)), and place the other two squares using their known orientations and locations relative to S1.
- Write equations for the interior lines that form the triangle. Each square contributes an inner edge (a line segment) that bounds the central triangular gap; those edges are straight lines whose positions can be written explicitly from the coordinates of the square corners.
- Solve the three pairs of line equations to find the three intersection points (the triangle’s vertices).
- Compute the Euclidean distance between the intersection points to get the three side lengths of the central triangle.

(If you prefer vector algebra: represent each inner edge as a line given by a point plus a direction vector, compute intersection points by solving two linear equations, then take vector differences to get side lengths.)

If you give me the actual side lengths and a sketch or the coordinates/orientations, I can carry out these steps and give exact numbers.

B. Relationships among the areas of the squares
- Each square’s area is the square of its side length: area(Si) = si^2. So the areas are directly proportional to the squares of their side lengths.
- If the geometry is symmetric (for example three congruent squares arranged symmetrically), the central triangle will inherit symmetry (e.g., be equilateral or isosceles) and the areas of the three squares will be equal.
- More generally, if the triangle side lengths can be expressed as simple linear combinations of the square side lengths (this happens in common arrangements), then area relations among the squares can be converted into relations among the triangle sides by squaring those linear combinations.
- Important scaling fact: if you scale every square side by the same factor k, each square’s area scales by k^2 and every linear dimension of the central triangle scales by k (so the triangle’s area scales by k^2 as well).

C. How to make each recreation area smaller while keeping a similar central triangle
- Scale every square by the same factor k with 0<k<1 and keep their orientations and relative positions exactly the same. Concretely, multiply every side length (s1, s2, s3) and every coordinate by k. Because similarity is preserved by uniform scaling, the central triangle will be similar to the original central triangle, but all areas (each square’s area and the triangle’s area) will be multiplied by k^2, so each recreation area becomes smaller by that factor.
- If you want a specific target (for example each square’s area reduced to 60% of its original), choose k = sqrt(0.60) and scale all sides and coordinates by that k.
- If you scale only some squares and not others, the central triangle will usually change shape (it will not be similar to the original), so scale all three equally to keep similarity.

If you can upload the sketch or give the side lengths and how the three squares meet, I’ll compute the exact triangle side lengths and the numeric area relationships.

make it simple

Answer 1

Here is a much simpler, plain-language version of A–C.

A. Finding the three side lengths of the central triangle
- Label the square side lengths s1, s2, s3 and fix how the squares sit next to each other.
- Put the picture on coordinates (for example put one square along the x‑axis with corners at (0,0) and (s1,0)). Place the other squares using their known positions/orientations.
- Each square contributes one straight inner edge that borders the central gap. Write the equations (or parametric forms) of those three lines from the square corner coordinates.
- The triangle vertices are the three pairwise intersections of those lines. Solve two line equations at a time to get the three vertex coordinates.
- The triangle side lengths are the distances between those vertices (Euclidean distance).

If you prefer vector form: represent each edge as point + direction vector, solve two linear equations for intersections, then take vector differences to get lengths.

Give me the square side lengths and how they touch or a sketch and I’ll compute the numbers.

B. Simple area relationships
- Area of square Si = si^2.
- If you scale all three squares by the same factor k, every square’s area multiplies by k^2 and every linear length (including the triangle sides) multiplies by k.
- If the squares are congruent and symmetrically placed, their areas are equal and the central triangle may be regular (e.g., equilateral). Otherwise areas follow directly from s1^2, s2^2, s3^2.

C. Making all recreation areas smaller but keeping the same triangle shape
- Scale every square and every coordinate by the same factor k (0 < k < 1). That makes the central triangle similar to the original but smaller.
- Areas shrink by k^2. Example: to make each square’s area 60% of original, choose k = sqrt(0.60) ≈ 0.7746 and scale all side lengths by that k.
- Do not scale only some squares if you want to preserve the triangle’s shape — unequal scaling changes the triangle.

If you want, send the side lengths or a photo/sketch and I’ll compute the exact triangle side lengths and areas.