diff --git "a/HarvardMIT/md/en-282-2025-feb-geo-solutions.md" "b/HarvardMIT/md/en-282-2025-feb-geo-solutions.md"
--- "a/HarvardMIT/md/en-282-2025-feb-geo-solutions.md"
+++ "b/HarvardMIT/md/en-282-2025-feb-geo-solutions.md"
@@ -13,7 +13,7 @@ Answer: 26
 
 Solution:  
 
-![](data:image/jpeg;base64,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LzySPkT55XxyVVRuIHJ4HfNdbXM+ItIupfEWga/aQG5bS2nSS3VgGaOVApZSSBuUqpwSMjPOcZBkl/pOp22hXU9lql5ca0kDtDJIw2SSBSQvlDCBSeOmffPNbdlFLb2NvDPMZpo4lR5W6uwGCx+p5qn9ovL+aJYbee0t1cPLLKArOAc7FXJPJ6k446ZzkadABRRRQAUUUUAFFFFABRRRQAUUUUAFY3iXw5D4o03+z7q9u7e2Lo8iW/ljzNrBgG3I3GVHAxmiigBLjw/JeSQfatb1KaCKVZWtyIVSUqcgPtjDEZAOARnHORW1RRQAUUUUAFFFFABRRRQAUUUUAFYcXhmOLxVJ4hOpX0l1JALbyn8ryxEGLBQAmRyeucnjJoooA3KKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigD//Z)
+![md5:a6173614c1296050c66bb92f443fe679](a6173614c1296050c66bb92f443fe679.jpeg)
   
 
 Extend \(DE\) to meet \(AC\) at \(X\) . Observe that \(ABEX\) and \(DFCX\) are isosceles trapezoids (both with base angles of \(60^{\circ}\) ), so we have  
@@ -33,7 +33,7 @@ Answer: 63
 Solution: Note that the difference in heights between the two stalactites does not equal the difference in heights between the stalagmites. This tells us that the two stalactites form a pair of opposite vertices of the square, and likewise for the stalagmites. As the midpoint of the pairs of structures must then coincide, we know that it is \(\frac{16 + 36}{2} = 26\) from the ceiling due to the stalactites, and \(\frac{25 + 49}{2} = 37\) from the ground due to the stalagmites. Therefore the height of the cave is just the sum of these two values, i.e. 63.
 
 
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+![md5:fd62624cfb0627a2adc3480c4bc66874](fd62624cfb0627a2adc3480c4bc66874.jpeg)
   
 
 3. Point \(P\) lies inside square \(ABCD\) such that the areas of \(\triangle PAB\) , \(\triangle PBC\) , \(\triangle PCD\) , and \(\triangle PDA\) are 1, 2, 3, and 4, in some order. Compute \(PA \cdot PB \cdot PC \cdot PD\) .  
@@ -44,7 +44,7 @@ Answer: \(\boxed{8\sqrt{10}}\)
 
 Solution:  
 
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+![md5:f473be8f087c94b0509ca94aee145a54](f473be8f087c94b0509ca94aee145a54.jpeg)
   
 
 Let \(h_{1}\) , \(h_{2}\) , \(h_{3}\) , and \(h_{4}\) be the lengths of the altitudes from \(P\) to sides \(AB\) , \(BC\) , \(CD\) , and \(DA\) , respectively. Then, the problem statement implies that \(\{h_{1}, h_{2}, h_{3}, h_{4}\} = \{x, 2x, 3x, 4x\}\) for some \(x\) . Furthermore, the area of the square is \(1 + 2 + 3 + 4 = 10\) , so we have  
@@ -61,7 +61,7 @@ Hence either \((\{h_{1}, h_{3}\} = \{x, 4x\}\) and \(\{h_{2}, h_{4}\} = \{2x, 3x
 
 Semicircle \(S_{1}\) is inscribed in a semicircle \(S_{2}\) , which is inscribed in another semicircle \(S_{3}\) . The radii of \(S_{1}\) and \(S_{3}\) are 1 and 10, respectively, and the diameters of \(S_{1}\) and \(S_{3}\) are parallel. The endpoints of the diameter of \(S_{3}\) are \(A\) and \(B\) , and \(S_{2}\) 's arc is tangent to \(AB\) at \(C\) . Compute \(AC \cdot CB\) .  
 
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)
+![md5:6fc5d1b7698693176f0318294751a3ba](6fc5d1b7698693176f0318294751a3ba.jpeg)
   
 
 Proposed by: Karthik Venkata Vedula  
@@ -70,7 +70,7 @@ Answer: 20
 
 Solution:  
 
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+![md5:663200b55e5c280cca76a457c7d2b8f7](663200b55e5c280cca76a457c7d2b8f7.jpeg)
   
 
 Let \(P\) , \(Q\) , and \(R\) be the midpoints of the diameters (i.e., the center of the circular arcs) of \(S_{3}\) , \(S_{2}\) , and \(S_{1}\) , respectively. Observe that if one fixes \(S_{3}\) , the location of \(S_{2}\) is uniquely determined by the angle between the diameters of \(S_{2}\) and \(S_{3}\) . The same holds for \(S_{2}\) and \(S_{1}\) . Thus, the figures \(S_{3} \cup S_{2}\) and \(S_{2} \cup S_{1}\) are similar. This gives us that the radius of \(S_{2}\) is \(\sqrt{10}\) .  
@@ -86,7 +86,7 @@ Answer: \(\sqrt{23} - 2\sqrt{3}\)
 Solution:
 
 
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+![md5:bd8399251a351972a45c43c96e817f24](bd8399251a351972a45c43c96e817f24.jpeg)
   
 
 Let \(A^{\prime}\) be the antipode of \(A\) . As \(\angle A X A^{\prime} = 90^{\circ}\) , we have \(X\) , \(P\) , and \(A^{\prime}\) are collinear. As \(\angle B P C = 120^{\circ}\) and \(\angle B A^{\prime}C = 180^{\circ} - \angle B A C = 120^{\circ}\) , it follows that \(P\) lies on the circle with center \(A^{\prime}\) passing through \(B\) and \(C\) , so \(A^{\prime}P = \frac{B C}{\sqrt{3}} = 2\sqrt{3}\) and \(A A^{\prime} = 2A^{\prime}B = 4\sqrt{3}\) . By the Pythagorean theorem, \(X A^{\prime} = \sqrt{(4\sqrt{3})^{2} - 5^{2}} = \sqrt{23}\) , so the answer is \(X A^{\prime} - P A^{\prime} = \left[\sqrt{23} - 2\sqrt{3}\right]\) .  
@@ -99,7 +99,7 @@ Answer: \(\boxed{9\sqrt{15}}\)
 
 Solution:  
 
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+![md5:b89478beaacda454384b1b3adf2b03df](b89478beaacda454384b1b3adf2b03df.jpeg)
   
 
 Let \(O_{A}\) , \(O_{B}\) , \(O_{C}\) , and \(O_{D}\) be the circumcenters of \(\triangle B C D\) , \(\triangle C D A\) , \(\triangle D A B\) , and \(\triangle A B C\) , respectively. Note that \(O_{B}O_{C}\) is the perpendicular bisector of \(\overline{{A D}}\) . Similarly, \(O_{B}O_{D}\perp A C\) and \(O_{C}O_{D}\perp A B\) . As \(A B\parallel C D\) , we have \(O_{C}O_{D}\perp C D\) . Then, \(\triangle O_{B}O_{C}O_{D}\stackrel {\star}{\sim}\triangle A D C\) , as their corresponding sides are perpendicular. Likewise, \(\triangle O_{D}O_{A}O_{B}\stackrel {\star}{\sim}\triangle C B A\) , so \(O_{A}O_{B}O_{C}O_{D}\stackrel {\star}{\sim}B A D C\) .  
@@ -126,7 +126,7 @@ Answer: \(\frac{7}{18}\)
 
 Solution 1:  
 
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+![md5:0f8d285c7902e9de79a6c8453e3fe343](0f8d285c7902e9de79a6c8453e3fe343.jpeg)
   
 
 Let the internal and external bisectors of \(\angle B A C\) meet \(B C\) at \(D\) and \(E\) . Similarly, let the internal and external bisectors of \(\angle B P C\) meet \(B C\) at \(D^{\prime}\) and \(E^{\prime}\) . The angle condition implies that \(A D\parallel P D^{\prime}\) and \(A E\parallel P E^{\prime}\) . Thus, triangles \(A D E\) and \(P D^{\prime}E^{\prime}\) are homothetic. Hence, the requested ratio is \(\frac{D E}{D^{\prime}E^{\prime}}\) . Repeated applications of the angle bisector theorem yield  
@@ -138,7 +138,7 @@ so \(D E = 24\) and \(D^{\prime}E^{\prime} = 28 / 3\) . Hence, the answer is \(\
 Solution 2:
 
 
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+![md5:81c307a05321f49518a124cd36d44fda](81c307a05321f49518a124cd36d44fda.jpeg)
   
 
 Let \(D = A P\cap B C\) \(E = B P\cap A C\) and \(F = C P\cap A B\) . Then, \(B C E F\) is cyclic, so by Power of a Point, \(\frac{A E}{A F} = \frac{A B}{A C} = \frac{3}{4}\) . Let \(A E = 3x\) and \(A F = 4x\) . Then,  
@@ -175,7 +175,7 @@ Answer: \(\sqrt{6}\)
 
 Solution:  
 
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+![md5:99272909b8e7b4116af394fdccc3eb87](99272909b8e7b4116af394fdccc3eb87.jpeg)
   
 
 Let \(D^{\prime}\) be the reflection of \(D\) across \(M\) . Then, \(A D E D^{\prime}\) is a parallelogram. Hence, \(D^{\prime}A = 2\) , so \(D^{\prime}B = 6\) . Thus, if \(D^{\prime}M = M D = x\) , then Power of a Point at \(D^{\prime}\) gives \(x\cdot (2x) = 2\cdot 6\) , so \(x = \sqrt{6}\) .  
@@ -197,12 +197,12 @@ Let \(P_{1}\) and \(P_{2}\) be the projections of \(O_{1}\) and \(O_{2}\) onto s
 Solving this, we get \(x = 7 + 2\sqrt{37}\) , which implies that \(A B = 2x = \boxed{14 + 4\sqrt{37}}\) . (The condition that \(\triangle A X D\) and \(\triangle B X C\) are acute rules out \(14 - 4\sqrt{37}\) .)
 
 
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+![md5:73c875021da9d1a19da0890bf095a65e](73c875021da9d1a19da0890bf095a65e.jpeg)
   
 
 Solution 2:  
 
-![](data:image/jpeg;base64,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)
+![md5:666d9c083cfaee5f174d61d41f0513fe](666d9c083cfaee5f174d61d41f0513fe.jpeg)
   
 
 Let \(P\) be the antipode of \(A\) in \(\odot (AXD)\) and \(Q\) be the antipode of \(B\) in \(\odot (BXC)\) . From \(\angle PDA = \angle QCB = 90^{\circ}\) , we get that \(P\) and \(Q\) lie on \(CD\) . Moreover, from \(\angle PXA = 90^{\circ}\) , we get that \(P \in BX\) , and similarly \(Q \in AX\) .  
@@ -226,7 +226,7 @@ from which it follows that \(t_{a} = 5\) , \(t_{b} = 6\) , and \(t_{c} = 7\) .
 Now, a translation of a plane can be written in one of three equivalent forms: it can be viewed as a translation in the \(x\) direction by a distance \(d_{x}\) , a translation in the \(y\) direction by a distance \(d_{y}\) , or a translation in the \(z\) direction by a distance \(d_{z}\) (with coordinate axes chosen as shown below).
 
 
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+![md5:aa24b6ffd361094bfb5582485f15d72d](aa24b6ffd361094bfb5582485f15d72d.jpeg)
   
 
 As shown above, we can express \(t_{a}\) , \(t_{b}\) , and \(t_{c}\) in terms of \(d_{x}\) , \(d_{y}\) , and \(d_{z}\) using the Pythagorean theorem, which yields \(t_{a} = \sqrt{d_{y}^{2} + d_{z}^{2}}\) , \(t_{b} = \sqrt{d_{z}^{2} + d_{x}^{2}}\) , and \(t_{c} = \sqrt{d_{x}^{2} + d_{y}^{2}}\) . Hence,  
@@ -239,7 +239,7 @@ We can draw a right pyramid with legs \(d_{x}\) , \(d_{y}\) , and \(d_{z}\) whic
 
 Solution 2: Let the vertices of the hexagon be \(ABCDEF\) , where \(AB = 45\) , \(BC = 66\) , etc. Note that \(AB \parallel DE\) , \(BC \parallel EF\) , and \(CD \parallel FA\) . Let \(O\) be the center of the prism, and let \(M\) , \(N\) , and \(P\) be the midpoints of \(AD\) , \(BE\) , and \(CF\) , respectively.  
 
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