There is no such city. The numbers do not work out.
Doha, the capital city of Qatar, is located 720 miles south of Tehran, Iran and 300 miles east of Riyadh, Saudi Arabia.
It is Mecca, Saudi Arabia.
None. Saudi Arabia is the only country that borders Egypt to the east besides Israel. Jerusalem is too close to Cairo to be considered for this question, and is also north of Riyadh. Riyadh itself is 1000 miles southeast of Cairo, so it makes no sense that there would be a capital city equidistant from the two. Sudan Borders Egypt to the south, but its capital Khartoum is 1000 miles from Cairo. However, the capitals of Yemen and Oman, both south of riyadh (saudi arabia) are roughly 720 miles from riyadh. However, they do not meet the requirements for this question, as they are 1300 and 1700 miles from Cairo respectively.
The ARABIAN SEA is 1700 miles southeast of Riyadh, Saudi Arabia.
The driving distance from Riyadh, Saudi Arabia to Cairo, Egypt is 1,724 miles / 2,775 km
That point is in the southeast Libyan desert, roughly 900 miles southeast of Tripoli, and 700 miles southwest of Cairo, Egypt.
Riyadh is 270 miles southeast of Bahrain.
The capital that is approximately 270 miles southwest of Bahrain and 525 miles east of the Red Sea is Riyadh, the capital of Saudi Arabia.
Riyadh
The capital city located 270 miles southwest of Bahrain and 525 miles east of the Red Sea is Riyadh, the capital of Saudi Arabia.
That point is in northeastern Egypt, about 87 miles southeast of the center of Cairo, and about 74 miles southwest of the southern entrance to the Suez Canal.
Riyadh, Saudi Arabia
Well the answer is somewhere
That point is in southeastern Egypt, roughly 58 miles west-southwest of Aswan, 138 miles north of the border with Sudan, and 420 miles south-southeast of the center of Cairo.