Largest Quadrilateral inscribed in Scalene Triangle

[Zane1]

I have a scalene triangle:
A: 75.04
B: 66.9
C: 41.13

The first thing I need to do is move just lines A and C in towards each other .5 and recalculate all sides.

Then I need to inscribe the largest quadrilateral that will fit while having one side being no shorter than 7.5, with the entire quadrilateral maintaining a distance of 5 away from all three sides of the scalene triangle.

I need the new measurements for the scalene triangle and the quadrilateral.

 

[jvicens]

Hello,

I would like to offer some suggestions for your problem.

Firstly, to move lines A and C towards each other by 0.5, we can use the concept of congruent triangles. We can draw a line perpendicular to line A, intersecting at point D, and another line perpendicular to line C, intersecting at point E. This will create two right triangles, ADE and CDE, which will be congruent to each other. By moving point D and E towards each other by 0.5, we can achieve the desired result of reducing the lengths of lines A and C by 0.5.

Next, to find the new measurements for the scalene triangle, we can use the Pythagorean Theorem to calculate the length of the third side (let's call it line B') of the triangle. We know that the sum of the squares of the two shorter sides of a right triangle is equal to the square of the longest side. So, we can set up the following equation:

(A')^2 + (B')^2 = (C')^2

Where A' and C' are the new lengths of lines A and C, and B' is the length of the new line B. Solving for B', we get B' = √[(C')^2 - (A')^2].

Similarly, we can use the Pythagorean Theorem to calculate the length of the new line B' for the quadrilateral. Since we want the quadrilateral to have a minimum side length of 7.5, we can set up the following equation:

(B')^2 + (C')^2 = (7.5)^2

Solving for B', we get B' = √[(7.5)^2 - (C')^2].

Finally, to maintain a distance of 5 away from all three sides of the scalene triangle, we can use the concept of parallel lines. We can draw a line parallel to line A, and another line parallel to line C, both at a distance of 5 away from lines A and C respectively. These parallel lines will intersect at point F, creating a rectangle with sides of 5 and (C'-5). By calculating the length of the diagonal of this rectangle, we can find the length of the fourth side of the quadrilateral (let's call it line D