Can AI handle geometry problems?

[mathman]

TL;DR

Can AI such as GPT-4 handle a geometry problem?

Can AI or GPT-4 answer the following?
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Geometry problem: Triangle with known length sides a, b, c where a is the longest. Place inside the triangle a semi-circle with diameter on side a. What is radius of largest possible semi-circle in terms of side lengths?

 

[jedishrfu]

I applied it to CHatGPT using your geometry problem text as the query and got the following:

Let's start by drawing the triangle and the semicircle.

non-existent url to an image of a triangle

As given, the diameter of the semicircle is equal to the longest side of the triangle, which is a. Let O be the center of the semicircle, and let D be the midpoint of side a.

non-existent url to an image of a triangle with mid point D

Since OD is the radius of the semicircle, we want to find its maximum value. To do this, we need to find the largest possible value of OD such that the semicircle still fits inside the triangle ABC.

Since a is the longest side of the triangle, we can draw a perpendicular from O to side a at point D. Let's call this point E.

non-existent url to an image of a triangle with perpendicular

Notice that OE is the radius of the circle, and DE is half of side a. So we want to maximize OE while keeping it less than or equal to DE.

Using the Pythagorean theorem in triangle ODE, we have:

OE^2 = OD^2 - DE^2

Substituting OD = a/2 and DE = a/2, we get:

OE^2 = (a/2)^2 - (a/2)^2 = 0

This tells us that OE = 0, which means that the largest possible semicircle is just a line segment. Therefore, the radius of the largest possible semicircle is 0.

In other words, it is not possible to place a semicircle with a positive radius inside the triangle ABC with diameter on side a.

What do you think? Did ChatGPT get it right? or what did it get wrong?

ChatGPT and other LLM apps basically start with a seed prompt and then generate the next word followed by the next word...

Consequently, it may answer correctly but its reasoning is all wrong or the reasoning is right but the computed answer is wrong or the citations and URLS it gives are either made up or non-existent or may not correspond to the problem being solved.

Use ChatGPT with a buyer beware attitude (caveat emptor)!

 

[mathman]

This initial statement is incorrect. Diameter lies on a, but its length is unknown.

"As given, the diameter of the semicircle is equal to the longest side of the triangle, which is a"

 

[mathman]

Corrected:

 

[mathman]

Revise to clarify.

Geometry problem: Semi-circle inside triangle: Triangle with known length sides a, b, c where a is the longest. Place inside the triangle a semi-circle with diameter on side a. What is radius of largest possible semi-circle in terms of side lengths? Position of diameter center along a?

 

[Tom.G]

Without doing the math, consider this special case:

Visualize a triangle with the Base being a horizontal line.
Side 'a' extends upward at some unknown angle from the right end of the Base.
Point 'D' is at the midpoint of side 'a'.

1. For maximum radius, the original Triangle must be equilateral. (any other shape will have a single, more acute angle, at one end of line 'a', thereby forcing the Arc outside the Triangle)
2. Drop a vertical from point 'a' to the Base.
3. This vertical yields a right angle at the Base.
4. The two angles at the base are 90° and 60° ( per step 1. and steps 2.&3.)
4a. The remaining angle, where the vertical meets side 'a', is 30°
5. Therefore, the length of the vertical from the Base to point 'D', distance 'x', is

Cos(30) = 'x' / ('a'/2)
-or-
'x' = Cos(30) × ('a'/2)
​

'x' is the maximum arc radius to keep the arc within the triangle.

Generalizing this to an arbitrary triangle is left as an exercise for the reader.

Looking forward to that generalization,
Tom

 

[mathman]

Your special case is not the same as the given problem. Original problem - the three sides are prescribed and must be used as is. The question is for any given triangle. I am asking if AI can find the general solution.

 

[Halc]

1. For maximum radius, the original Triangle must be equilateral. (any other shape will have a single, more acute angle, at one end of line 'a', thereby forcing the Arc outside the Triangle)

Well I agree that it must be equilateral, but not for the reason given. An asymmetric triangle would have the inscribed semicircle off center is all.

For a given length a, the longer the other two sides are, the larger the semicircle, so given the constrain that b and c are no longer than a, the optimal solution seems to be an equilateral triangle. Trick now is to get the GPT to work that out.

It seems to be partially an unclear problem description.

Geometry problem: Triangle with known length sides a, b, c where a is the longest. Place inside the triangle a semi-circle with diameter on side a. What is radius of largest possible semi-circle in terms of side lengths?

This doesn't say 'for a given length 'a'. It seems to leave all of them open, like perhaps it's asking for the largest semicircle-radius to triange-perimeter-length ratio, which would be closer to a right triangle.

 

[TonyStewart]

ChatGPT3.5 cannot seem to solve this geometry even when I told it to assume b=c after a few tries.

To make the semi-circle fit inside the triangle, the diameter, d is almost = a while the triangle height is infinite so that the tangent at 0 and 180 deg is almost 90 deg. thus the ratio of a/b =0 with b= infinity and Diameter = a

r = (a/2) * \sqrt{{a^2 - b^2 - c^2}^2 - 4b^2c^2} / {a^2 - b^2 - c^2}

 

[mathman]

ChatGPT3.5 cannot seem to solve this geometry even when I told it to assume b=c after a few tries.

To make the semi-circle fit inside the triangle, the diameter, d is almost = a while the triangle height is infinite so that the tangent at 0 and 180 deg is almost 90 deg. thus the ratio of a/b =0 with b= infinity and Diameter = a

r = (a/2) * \sqrt{{a^2 - b^2 - c^2}^2 - 4b^2c^2} / {a^2 - b^2 - c^2}

The side lengths are given. For example a=4,b=3,c=2. Find that semi-circle.

 

[mathman]

Tried it on a GPT. Got no response!

 

[TonyStewart]

The side lengths are given. For example a=4,b=3,c=2. Find that semi-circle.

I have found ChatGPT has errors in computing formulae with division. This took 3 tries and I wouldn't trust it.

3 / 3
The formula provided is for the radius of the circle circumscribing a triangle, given the lengths of its sides a, b, and c. The formula is:
r = (a/2) * √((a^2 - b^2 - c^2)^2 - 4b^2c^2) / (a^2 - b^2 - c^2)
Using the values a = 4, b = 3, and c = 2, we can substitute them into the formula:
r = (4/2) * √((4^2 - 3^2 - 2^2)^2 - 4(3^2)(2^2)) / (4^2 - 3^2 - 2^2) r = 2 * √(7^2 - 36) / 7 r = 2 * √13 / 7
This is the radius of the circle circumscribing the triangle. To find the area of the semicircle, we need to use the formula for the area of a circle:
A = πr^2
The area of the semicircle is half of this:
A(semicircle) = 1/2 * π * r^2 A(semicircle) = 1/2 * π * (2 * √13 / 7)^2 A(semicircle) = π * 52 / 49
Therefore, the area of the semicircle is π * 52 / 49 square units.

 

[mathman]

The semi-circle is inside the triangle. I tried chatgpt and got no answer. I also asked for pthag. triplets for c < 50 and got some wrong answers.

 

[TonyStewart]

By re-writing the question, ChatGPT got an answer for the Diameter using Heron's formula and assuming c = triangle base for a,b,c sides ,where the height , h = Diameter of largest circle.
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make sure you state the largest side is the base = cApologies for the oversight. If we assume that side c is the largest side of the triangle and we want to consider it as the base, we can proceed as follows:
To find the area and base height of a triangle with sides a, b, and c, where c is the largest side and considered the base, you can use the following steps:

1. Determine the semiperimeter (s) of the triangle: s = (a + b + c) / 2
2. Calculate the area (A) of the triangle using Heron's formula: A = sqrt(s * (s - a) * (s - b) * (s - c))
3. Calculate the base height (h) corresponding to side c: h = 2A / c

By applying these formulas, you can find the area and base height of the triangle with sides a, b, and c, where c is considered the largest side and used as the base. Remember to ensure that the given side lengths form a valid triangle, where the sum of any two sides is greater than the third side.

 

[mathman]

By re-writing the question, ChatGPT got an answer for the Diameter using Heron's formula and assuming c = triangle base for a,b,c sides ,where the height , h = Diameter of largest circle.
====
make sure you state the largest side is the base = cApologies for the oversight. If we assume that side c is the largest side of the triangle and we want to consider it as the base, we can proceed as follows:
To find the area and base height of a triangle with sides a, b, and c, where c is the largest side and considered the base, you can use the following steps:

1. Determine the semiperimeter (s) of the triangle: s = (a + b + c) / 2
2. Calculate the area (A) of the triangle using Heron's formula: A = sqrt(s * (s - a) * (s - b) * (s - c))
3. Calculate the base height (h) corresponding to side c: h = 2A / c

By applying these formulas, you can find the area and base height of the triangle with sides a, b, and c, where c is considered the largest side and used as the base. Remember to ensure that the given side lengths form a valid triangle, where the sum of any two sides is greater than the third side.

You state that chatgot gave an answer, but your description us for the triangle area and not for the semicircle inside.

 

[TonyStewart]

You state that chatgot gave an answer, but your description us for the triangle area and not for the semicircle inside.

That’s because I knew the height was the answer needed

 

[mathman]

Latest from chatgpt using:
Geometry problem: Semi-circle inside triangle: Triangle with known length sides a, b, c where a is the longest. Place inside the triangle a semi-circle with diameter resting on side a. What is radius of largest possible semi-circle in terms of side lengths? Position of diameter center along a?
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The diameter center is at a/2 and the radius is ab/(2c) - obviously wrong. It seems to be unable to get the diagram correctly.

 

[TonyStewart]

Latest from chatgpt using:
Geometry problem: Semi-circle inside triangle: Triangle with known length sides a, b, c where a is the longest. Place inside the triangle a semi-circle with diameter resting on side a. What is radius of largest possible semi-circle in terms of side lengths? Position of diameter center along a?
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The diameter center is at a/2 and the radius is ab/(2c) - obviously wrong. It seems to be unable to get the diagram correctly.

Both your statements are false and radius=r=h
r= h=2A / c
A = sqrt(s * (s - a) * (s - b) * (s - c))
s = (a + b + c) / 2
for abc=3,4,5, s=6, A=6, r=12/5=2.4

If you want a to be longest swap a to c

 

[mathman]

Both your statements are false and radius=r=h
r= h=2A / c
A = sqrt(s * (s - a) * (s - b) * (s - c))
s = (a + b + c) / 2
for abc=3,4,5, s=6, A=6, r=12/5=2.4

If you want a to be longest swap a to c

What I gave was the wrong answer from chatgpt not my answer.

My answer is r=12/7 and the center at 20/7 along long side measured from corner opp. side 4.

 

[TonyStewart]

Maybe we all got it wrong ( incl open.AI)
Using my simple logic, using 2 limit cases;

1. The largest semi-circle radius, R to fit inside any triangle for any size;
- R cannot exceed the smallest side,
- for an extremely small side, it would be approximately equal to R

2. With 2 equal sides, in a right angle triangle, the diameter = twin side, R=side/2
thus R=0.5b=0.5c=0.707a (0.707=sqrt2/2)

This for any triangle the radius R MUST be between 50 & 100% of the smallest side
for lengths=3,4,5, I said R was 2.4 or 80% of the smallest side
you said 1.71=12/7 or 57% of the smallest side, which doesn't sound right for a right angle triangle.

(I measured it closer to 80.8% of the smallest side =3)

 

[TonyStewart]

I still got it wrong, for a triangle with an extremely small angle and opposite side, that will be the semi-circle diameter, D thus the smallest R is 50% of the smallest side.

Yet somehow R= h to apex corner, predicted by OpenAI seem to predict a reasonably accurate result.

R= h=2*Area / largest side

Area = A = sqrt(s * (s - a) * (s - b) * (s - c))
s = (a + b + c) / 2

 

[mathman]

My answers are $r=\frac{b\times c}{b+c}sin(A)$ and,$x=\frac{a\times c}{b+c},\ x$ measured from angle B

I get r as the distance from the diameter center to both other sides points of tangency. I use sine law to get rid of sin(B) and sin(C). AI radius is not radius of semicircle.